Posted October 14, 2003

Last modified October 1, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Tuesday, September 23, 2003 Lockett 240
Stephen Shipman, Mathematics Department, LSU

Boundary projections and Helmholtz resonances 1

Posted October 14, 2003

Last modified October 1, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Tuesday, September 30, 2003 Lockett 240
Stephen Shipman, Mathematics Department, LSU

Boundary projections and Helmholtz resonances 2

Posted October 23, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Harris Wong, Department of Mechanical Engineering

A d-function model of facets and its applications

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Stephanos Venakides, Department of Mathematics, Duke University

The Semiclassical Limit of the Focusing Nonlinear Schroedinger Equation

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Robert Lipton, Mathematics Department, LSU

Field Fluctuations, Spectral Measures, and Moment Problems

Posted October 24, 2003

Last modified November 6, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Christo Christov, University of Louisiana at Lafayette

Nonlinear Waves and Quasi-Particles: The Emerging of a New Paradigm

Posted October 24, 2003

Last modified November 6, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Karsten Thompson, Department of Chemical Engineering, Louisiana State University

Modeling Multiple-scale Phenomena in Porous Materials

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Endel Iarve, Materials Directorate Wright Patterson Air Force Base and the University of Dayton Research Institute, Dayton Ohio

Mesh-independent modeling of cracks by using higher-order shape functions

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm The Deans conference room 3225, CEBA
Endel Iarve, Materials Directorate Wright Patterson Air Force Base and the University of Dayton Research Institute, Dayton Ohio

Effect of splitting on tensile strength distribution of unidirectional carbon fiber composites

Special Civil Engineering and Applied Analysis Seminar

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

2:00 pm Lockett 285
Boris Baeumer, University of Otago, New Zealand

Fractal Transport and Dispersion: Limits of Continuous Time Random Walks

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Ricardo Estrada, Mathematics Department, LSU

Distributional Solutions of Singular Integral Equations

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Wilfrid Gangbo, Department of Mathematics, Georgia Institute of Technology

Inequalities for generalized entropy and optimal transportation

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Yitshak Ram, Department of Mechanical Engineering, Louisiana State University

Inverse Problems and Eigenvalue Assignment in Vibration and Control

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

Lockett 240
Yuri Antipov, Mathematics Department, LSU

Functional-difference equations and applications

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Jay Walton, Department of Mathematics, Texas A&M University

Dynamic Fracture Models in Viscoelasticity

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Manuel Tiglio, Department of Physics, Louisiana State University

Summation by parts and dissipation for black hole excision

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 240
Stephen McDowall, Department of Mathematics, Western Washington University Priklonsky

Total boundary determination of electromagnetic material parameters from boundary data

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall
Oscar Bruno, Department of Applied and Computational Mathematics, California Institute of Technology

New high-order, high-frequency methods in computational electromagnetism

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 240
Mayank Tyagi, Mechanical Engineering Department, Louisiana State University.

Issues in Large Eddy Simulations of Complex Turbulent Flows

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 240
Vladimir Priklonsky, Moscow State University

Tidal Flow and Transport Model

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 240
Paul Martin, Department of Mathematical and Computer Science, Colorado School of Mines, Golden

Fundamental solutions and functionally graded materials

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:00 pm Lockett Hall 240
Boris Belinskiy, Department of Mathematics, University of Tennessee at Chattanooga

Boundary Value Contact Problems

Posted October 24, 2003

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 240
Jannette Frandsen, Department of Civil & Environmental Engineering, LSU

A Tuned Liquid Damper Model for Frequency Response Predictions of a Coupled System

Posted August 20, 2003

Last modified September 17, 2003

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm Lockett 277
Gunter Lumer, University of Mons-Hainaut and Solvay Institute for Physics and Chemistry, Brussels

Multiparameter dynamics in macrophysics of clouds on flat and general surfaces, or in certain supply-management aspects

Posted September 4, 2003

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall
Jung-Han Kimn, Mathematics Department, LSU

Overlapping Domain Decomposition Methods

Posted September 4, 2003

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall
Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

Brittle fracture seen as a free discontinuities problem

Posted September 17, 2003

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall
Horst Beyer, Max Planck Institute for Gravitational Physics, Golm, Germany, and Dept. of Mathematics, LSU

On the Stability of the Kerr Black Hole

Posted September 23, 2003

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall
Olivier Sarbach, Dept. of Mathematics and Dept. of Physics & Astronomy, LSU

The initial-boundary value formulation of Einstein's equations

Posted August 14, 2003

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall
Gilles Francfort, Université Paris Nord, France

Brittle fracture evolution: a variational standpoint.

Posted August 14, 2003

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall
Yonggang Huang, Dept. of Mechanical Engineering, University of Illinois at Urbana-Champaign

The fundamental solution of intersonic crack propagation

Posted October 28, 2003

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 235, Lockett Hall
Andrej Cherkaev, University of Utah

TBA

To be followed by a \"Special Fluid Dynamic\" seminar at the Chimes.

Posted August 27, 2003

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett hall
Andri Gretarsson, California Institute of Technology and LIGO Livingston Observatory

Detecting Gravitational Waves

Posted October 23, 2003

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall Originally scheduled for Monday, October 27, 2003
Peter Y Huang, LSU, Department of Mechanical Engineering

Direct Numerical Simulation of Multiphase Flows in Newtonian and Non-Newtonian Fluids

Posted November 17, 2003

Applied Analysis Seminar Questions or comments?

4:00 pm - 5:00 pm 277, Lockett Hall
Darko Volkov, Department of Mathematical Sciences, New Jersey Institute of Technology

Integral equation methods for the statics and the dynamics of an
electrified fluid bridge

Posted January 8, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 277, Lockett Hall
Petr Kloucek, Computational and Applied Mathematics department, Rice University

Stochastic Modeling of the Functional Crystalline Materials

Posted January 22, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall Originally scheduled for Wednesday, February 11, 2004
Vladimir Mityushev, Institute de Physique du Globe de Paris (France), and Pedagogical University in Slupsk (Poland)

Effective properties of composites with unidirectional cylindrical fibers

Posted January 9, 2004

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall
Marcus Sarkis, Institito de Matematica Pura e Aplicada (IMPA, Brazil) and Worcester Polytechnic Institute

Schwarz Methods for Partial Differential Equations

Posted January 14, 2004

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm 235 Lockett Hall Originally scheduled for Monday, February 9, 2004
Ricardo Estrada, Mathematics Department, LSU

Distributional Radius of Curvature

We show that any continuous plane path that turns to the left has a well-defined distribution, that corresponds to the radius of curvature of smooth paths. As a byproduct, we will learn to divide by 0! These ideas were inspired by a talk by Professor H. Wong in the Applied Analysis Seminar some months ago, where he showed how to use Dirac delta functions to model facets in crystals.

Posted February 3, 2004

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

2:40 pm - 3:30 pm Lockett Hall 235
Gregory Kriegsmann, New Jersey Institute of Technology

Complete Transmission Through a Two-Dimensional Diffraction Grating

The propagation of a normally incident plane electromagnetic wave through a two-dimensional metallic grating, is modeled and analyzed using S-Matrix theory. The period of the structure $A$ is on the order of the incident wave length $\lambda$, but the height of the channel $H$ separating the grating elements is very small in comparison. Exploiting the small parameter $H/A$ an approximate transmission coefficient is obtained for the grating. For a fixed frequency this coefficient is $O(H/A)$ due to the thinness of the channel. However, near resonant lengths it is $O(1)$. That is, for certain widths the structure is transparent. Similarly, for a fixed length the transmission coefficient has the same resonant features as a function of frequency. This latter feature makes this grating potentially useful as a selective filter.

Posted February 11, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall
John Strain, University of California Berkeley

High-order fractional step methods for constrained differential equations

Posted April 12, 2004

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 235 Lockett Hall
Horst Beyer, Max Planck Institute for Gravitational Physics, Golm, Germany, and Dept. of Mathematics, LSU

On some vector analogues of Sturm-Liouville operators

The talk considers a general class of densely defined, linear symmetric operators in Hilbert space, which originate from the separation of vector partial differential operators (PDO) in three dimensions, which are invariant under the rotation group. Those PDO describe spheroidal Lagrangian adiabatic oscillations of spherically symmetric newtonian stars (treated as ideal fluids) in the so-called “Cowling approximation” in stellar pulsation theory. Their extension properties turn out to be very similar to that of minimal Sturm–Liouville operators. In particular close analogues of Weyl's famous theorems hold. On the other hand the spectral properties of their self-adjoint extensions are quite different. In particular every extension has a non-trivial essential spectrum. Finally, a result is given which allows to determine the resolvent of the self-adjoint extensions, which are perturbed by a “matrix” of integral operators of a specific general type. Those perturbed operators are generalizations of operators governing spheroidal adiabatic oscillations of spherically symmetric stars.

Posted January 9, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Friday, April 16, 2004 235, Lockett Hall
Guillermo Goldsztein , School of Mathematics, Georgia Institute of Technology

Perfectly plastic heterogeneous materials

Posted March 1, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall
Daniel Sage, Mathematics Department, LSU

Racah coefficients, subrepresentation semirings, and composite materials--An application of representation theory to material science

Posted April 26, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall
Helena Nussenzveig Lopes, Universidade Estadual de Campinas (Brasil) and Penn State University

On vortex sheet evolution

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents. LEQSF(2002-04)-ENH-TR-13

Posted September 10, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 239, Lockett Hall
Enrique Reyes, University of New Orleans

Practical finite difference modeling approaches to environmental problems: Louisiana coastal land loss.

Posted October 13, 2004

Applied Analysis Seminar Questions or comments?

3:30 pm 239, Lockett Hall
Stephen Shipman, Mathematics Department, LSU

Anomalous electromagnetic transmission mediated by guided modes

Posted October 26, 2004

Applied Analysis Seminar Questions or comments?

3:30 pm 239, Lockett Hall
Jonathan Dowling, Louisiana State University, Department of Physics
Hearne Professor of Theoretical Physics at LSU,
Quantum Sciences and Technologies Group

Effective densities of state

Posted October 26, 2004

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 239, Lockett Hall
Petr Plechak, Mathematics Institute, University of Warwick
Candidate for Associate Professor Position in Scientific Computation

TBA

Posted October 3, 2004

Last modified October 26, 2004

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm
Michel Jabbour, University of Kentucky

TBA

Posted October 27, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall
Asher Rubinstein, Department of Mechanical Engineering, Tulane University

Failure Analysis of Thermal Barrier Coatings

Posted October 26, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 239, Lockett Hall
Susanne Brenner, Department of Mathematics, University of South Carolina

Additive Multigrid Theory

Posted October 26, 2004

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall
Béatrice Rivière, Department of Mathematics, University of Pittsburgh

Discontinuous Galerkin methods for incompressible flows

Posted February 9, 2005

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:40 pm 239, Lockett Hall
Alexander Figotin, University of California at Irvine

Conservative extensions of dispersive dissipative systems

Posted March 31, 2005

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 285
John Willis, Cambridge University
Fellow, Royal Society of London (FRS)

Radon transforms in Solid Mechanics

Posted April 26, 2005

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:40 pm CEBA 2150
Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

From Geman and Geman to Mumford–Shah

This talk focuses on the issues raised by an apparently simple problem: extending Geman and Geman's weak-membrane model for the segmentation of signals to that of images. I will briefly describe the problems of image and signal segmentation, then present Geman and Geman's approach. I will illustrate the issue with its intuitive multi-dimensional extension. Then, I will present how one can derive the Mumford–Shah functional as the Gamma limit of a weak-membrane energy, and then extend it to the 2D and 3D cases. Time permitting, I will then present numerical schemes based on the Mumford–Shah problem.

Posted September 29, 2005

Last modified October 6, 2005

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 241 Lockett Hall
Robert Lipton, Mathematics Department, LSU

Multi-scale Stress Analysis

Many structures are hierarchical in nature and are made up of substructures distributed across several length scales. Examples include aircraft wings made from fiber reinforced laminates and naturally occurring structures like bone. From the perspective of failure initiation it is crucial to quantify the load transfer between length scales. The presence of geometrically induced stress or strain singularities at either the structural or substructural scale can have influence across length scales and initiate nonlinear phenomena that result in overall structural failure. In this presentation we examine load transfer between length scales for hierarchical structures when the substructure is known exactly or only in a statistical sense. New mathematical objects dubbed macrostress modulation functions are presented that facilitate a quantitative description of the load transfer in hierarchical structures. Several concrete physical examples are provided illustrating how these quantities can be used to quantify the stress and strain distribution inside multi-scale structures. It is then shown how to turn the problem around and use the macrostress modulation functions to design graded microstructures for control of local stress.

Posted October 11, 2005

Applied Analysis Seminar Questions or comments?

3:30 pm 241 Lockett Hall
Robert Lipton, Mathematics Department, LSU

Differentiation of G-limits and weak L-P estimates for sequences

Posted October 6, 2005

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm 241 Lockett Hall
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On the distance between homotopy classes of $S^1$-valued maps

Certain Sobolev spaces of $S^1$-valued functions can be written as a disjoint union of homotopy classes. The problem of finding the distance between different homotopy classes in such spaces is considered. In particular several types of one-dimensional and two-dimensional domains are studied. Lower bounds are derived for these distances. Furthermore, in many cases it is shown that the lower bounds are sharp but are not achieved.

Posted November 11, 2005

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm 239 Lockett Hall
Corey Redd, Department of Mathematics, LSU

Capturing Deviation from Ergodicity at Different Scales

Many researchers are interested in the topics of ergodicity and mixing, and more importantly in methods by which these quantities can be measured. As these properties may register differently based upon the space under observation, it is also important that any measure be able to be applied at different scales. Up to now, an energy based measure (L-2 norm) has traditionally been used to assess the ergodicity and/or mixing of a system. This method is less than ideal in part due to its non-uniqueness and difficulty with assessment on varying scales. I will present a Lagrangian based, multiscale method for measuring ergodicity that will attempt to address these issues. This talk will begin with background information on ergodicity and mixing and the relationship between the two. From the abstract definitions, I will derive an equation that will measure ergodicity on multiple scales. Following that, results will be presented from some initial computations of the metric on several test maps. Finally, computational issues will be discussed that are specific to measuring ergodicity, as well as in comparison to a mixing measure.

Posted January 23, 2006

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

12:30 pm - 1:30 pm 277 Lockett
Robert B. Haber, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign

Space-time Discontinuous Galerkin Methods for Multi-scale Solid Mechanics

Posted March 8, 2006

Last modified March 21, 2006

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Johnston 338
Ken Mattsson, Center for Integrated Turbulence Simulations, Stanford University

Towards time stable and high order accurate schemes for realistic applications

For wave propagation problems, the computational domain is often large compared to the wavelengths, which means that waves have to travel long distances during long times. As a result, high order accurate time marching methods, as well as efficient high order spatially accurate schemes (at least 3rd order) are required. Such schemes, although they might be G-K-S stable (convergence to the true solution as delta x -> 0), may exhibit a non-physical growth in time, for realistic mesh sizes. It is therefore important to device schemes, which do not allow a growth in time that is not called for by the differential equation. Such schemes are called strictly (or time) stable. We are particularly interested in efficient methods with a simple data structure that parallelize easily on structured grids. High order accurate finite difference methods fulfill these requirements. Traditionally, a successful marriage of high order accurate finite difference and strict stability was a complicated and highly problem dependent task, especially for realistic applications. The major breakthrough came with the construction (Kreiss et al., in 1974) of non-dissipative operators that satisfy a summation by parts (SBP) formulation, and later with the introduction of a specific procedure (Carpenter et al., in 1994) to impose boundary conditions as a penalty term, referred to as the Simultaneous Approximation Term (SAT) method. The combination of SBP and SAT naturally leads to strictly stable and high order accurate schemes for well-posed linear problems, on rectangular domains. During the last 10 years, the methodology has been extended to handle complex geometries and non-linear problems. In this talk I will introduce the original SBP and SAT concepts, and further discuss the status today and the focus on future applications. In particular I will discuss some recent developments towards time stable and accurate hybrid combinations of structured and unstructured SBP schemes, making use of the SAT method.

Posted January 26, 2006

Last modified March 11, 2006

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 235
Yaniv Almog, Department of Mathematics, LSU

Boundary layers in superconductivity and smectic liquid crystals

Posted April 13, 2006

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett, 235
Anna Zemlyanova, Department of Mathematics, LSU

The problem on reinforcement and repair of a hole in a plate with a patch

It is known that holes in a thin plate create undesirable stress concentration and can lead to the formation of cracks from the edge of the hole. I will consider the mechanical problem of repair of the hole by a two-dimensional patch. This problem will be reduced to the system of three singular integral equations. Uniqueness of the solution of the system will be proved. Numerical results will be given for some particular cases.

Posted May 2, 2006

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm Lockett 235
Gnana Bhaskar Tenali, Mathematics, Florida Institute of Technology

Fixed point theorems in partially ordered metric spaces and applications

I'll talk about some recent progress made on fixed point theorems in partially ordered metric spaces. In particular, I will discuss a fixed point theorem for a mixed monotone mapping in a metric space endowed with a partial order, using a weak contractivity type of assumption. Besides including several recent developments, such a theorem can be used to investigate a large class of problems. As an application we discuss the existence and uniqueness of solution for a periodic boundary value problem.

Posted June 20, 2006

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

2:30 pm - 3:30 pm 277, Lockett Hall
Mathias Stolpe, Institut for Mathematik, Danmarks Tekniske Universitet

A method for global optimization of the stacking sequence in laminated composite shell structures

Posted July 20, 2006

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 235, Lockett Hall
Fernando Fraternali, California Institute of Technology and Università di Salerno

Free Discontinuity Approaches to Fracture and Folding

Posted November 1, 2006

Last modified March 3, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 284 Lockett Hall
Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

Numerical implementation of a variational model of brittle fracture

Fracture mechanics is a very active area of research, with vital
applications. In recent years, the unexpected collapse of terminal 2F
at Charles de Gaulle airport in France or the Columbia space shuttle
disintegration upon re-entry illustrate the importance of a better
understanding and numerical simulation of the mechanism of fracture.

In the area of brittle fracture, the most widely accepted
theories are based on Griffith?s criterion and limited to the
propagation of an isolated, pre-existing crack along a given path.
Extending Griffith?s theory into a global minimization principle,
while preserving its essence, the concept of energy restitution in
between surface and bulk terms, G. Francfort and J.-J. Marigo
proposed a new formulation for the brittle fracture problem. The
basis of their model is the minimization of a total energy with
respect to any admissible displacement and crack field. The main
advantage of this approach is to be capable of predicting the
initiation of new cracks, computing their path, and accounting the
interactions between several cracks, in two and three space
dimensions. Of course, this has a price both theoretically and
numerically. In particular, in order to achieve global minimization
with respect to any crack set, one has to devise special numerical
methods.

After briefly reviewing the issues of brittle fracture mechanics,
I will present the Francfort-Marigo model. I will rapidly describe
some elements of its analysis, and present a numerical approximation
based on the properties of Gamma-convergence. I will derive necessary
optimality conditions with respect to the global time evolution, and
show how to use them in a minimization algorithm. Then, I will
present some extensions of the original model, accounting for body
forces (under some restrictions) or thermal loads, and describe how
to adapt the numerical implementation. I will illustrate my talk
with several large scale two and three dimensional experiments.

Posted November 1, 2006

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 284 Lockett Hall
Jung-Han Kimn, Mathematics Department, LSU

Parallel Implementation of Domain Decomposition Methods

Many important problems from current industrial and academic research, including the numerical solution of partial differential equations, generate extremely large data sets beyond the capacity of single-processor computers. Parallel computation on multiple-processor super computers is therefore the key to increasing performance but efficient parallel algorithms for multiple-processor super computers with huge number of processors are still needed. Domain Decomposition methods comprise an important class of parallel algorithms that are naturally parallel and flexible in their application to a sweeping range of scientific and engineering problems. This talk gives a brief discussion of some issues when we implement parallel domain decomposition methods. We will present some of our recent theoretical and numerical results for parallel domain decomposition methods for elliptic and hyperbolic partial differential equations.

Posted October 11, 2006

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

4:40 pm - 5:30 pm Room 284 Lockett Hall
Michael Mascagni, Department of Computer Science, Florida State University

Using Simple SDEs (Stochastic Differential Equations) to Solve Complicated PDEs (Partial Differential Equations)

This talk begins with an overview of methods to solve PDEs based on the representation of point solutions of the PDEs as expected values of functionals of stochastic processes defined by the Feynman–Kac formula. The particular stochastic processes that arise in the Feynman–Kac formula are solutions to specific SDEs defined by the characteristics of the differential operator in the PDE. The Feynman–Kac formula is applicable to wide class of linear initial and initial-boundary value problems for elliptic and parabolic PDEs. We then concentrate our attention on elliptic boundary value problems that arise in applications in materials science and biochemistry. These problems are similar in that the PDEs to be solved are rather simple, and hence the associated SDEs that arise in the Feynman–Kac formula are likewise simple. However, the geometry of the problem is often complicated and amenable to several acceleration approaches particular to these simple SDEs. We will specifically describe the walk on spheres, Greens function first passage, last passage, walk on the boundary, and walk on subdomains methods in this context. These methods will be presented in the setting of several applications studied by the author and his research collaborators.

Posted November 1, 2006

Last modified December 3, 2006

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 284
Robert Lipton, Mathematics Department, LSU

Homogenization and Field Concentrations in Heterogeneous Media

Posted November 1, 2006

Last modified December 11, 2006

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 284 Lockett Hall
Michael Stuebner, Louisiana State University

An inverse homogenization approach to stress constrained structural design

The presentation addresses the problem of optimal design of microstructure in composite materials. A computational method for grading the microstructure for the control of local stress in the vicinity of stress concentrations is developed. The method is based upon new rigorous multiscale stress criteria connecting the macroscopic or homogenized stress to local stress fluctuations at the scale of the microstructure. The approach is applied to different type of design problems.

Posted January 29, 2007

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

2:40 pm - 3:30 pm 338 Johnston Hall
Paul Saylor, University of Illinois

Stanford's Foresight and Forsythe's Stanford

What Stanford Was Like

What the Time Was Like

Over A Four Year Period

Starting with the Arrival of This New Man

Professor George Forsythe, In 1957

Plus A Bonus Look-Ahead to the Future

Posted January 29, 2007

Applied Analysis Seminar Questions or comments?

11:00 am - 12:00 pm Johnston Hall Room 338
Fengyan Li, Rensselaer Polytechnic Institute

Recent development in nonconforming methods for Maxwell equations

In this talk, I will discuss some recent developments in computational electromagnetism. Two schemes are formulated for the reduced time-harmonic Maxwell equations. One is using the classical nonconforming finite elements, the other is based on the interior penalty type discontinuous Galerkin methods. The operators in these schemes naturally define two Maxwell eigensolvers which are spurious free. Theoretical and numerical results will be presented to demonstrate the performance of these methods. This is joint work with Susanne Brenner and Li-yeng Sung (LSU).

Posted January 17, 2007

Last modified February 15, 2007

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 284 Lockett Hall
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On a minimization problem with a mass constraint involving a potential vanishing on two curves

We study a singular perturbation type minimization problem with a mass constraint over a domain or a manifold, involving a potential vanishing on two curves in the plane. We describe the asymptotic behavior of the energy as the parameter epsilon goes to zero, and in particular, how it depends on the geometry of the domain. In the case of the problem on the sphere we give a precise description of the limiting behavior of both the minimizers and their energies. This is a joint work with Nelly Andre (Tours).

Posted January 29, 2007

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 284 Lockett Hall
Alexander Pankov, College of William and Mary

Gap solitons, periodic NLS, and critical point theory

Here a gap soliton means a spatially exponentially localized standing wave solution of periodic nonlinear Maxwell equations, having a carrier frequency in a spectral gap. There is an enormous literature devoted to study of what should be gap solitons by means of approximate methods, e.g., envelope function approach, and numerical simulations (basically, in one dimension). These results provide a lot of information about such solutions, say, their shape. However, the existence of gap solitons is not a clear issue. In this talk we discuss the existence problem in the case of periodic Akhmediev-Kerr medium. We consider two-dimensional case and look for (TM) polarized solutions. Then the problem reduces to a (two-dimensional) periodic stationary NLS with cubic nonlinearity. To study this equation we employ critical point theory (specifically, the linking theorem) together with the so-called periodic approximations. This leads to the existence of TM gap solitons and provides an estimate for the rate of exponential decay. Finally, we discuss certain open mathematical problems.

Posted March 8, 2007

Last modified March 13, 2007

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 284 Lockett Hall
Hong Zhang, Dept. of Computer Science, Illinois Institute of Technology and Mathematics and Computer Science Division, Argonne National Laboratory

Eigenvalue Problems in Nanoscale Material Modeling

Together with a group of material scientist, we intend to calculate the atomic and electronic structure of nanoparticles on a quantum-mechanical level. The mathematical core of this modeling is a sequence of large and sparse eigenvalue problems. In this talk, I will present the special requirements of the solutions, the challenges on the computational method, our algorithmic approach and software development. Numerical implementation on the advanced distributed computers will be demonstrated.

This work also demonstrates how to efficiently develop special-purpose application code on the top of available parallel software packages. By the end of the talk, as a PETSc developer, I will give a demo on using PETSc (Portable, Extensible Toolkit for Scientific Computation) as a tool for large scale numerical simulation.

Posted February 12, 2007

Last modified April 23, 2007

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 284 Lockett Hall
Lia Bronsard, McMaster University

Ginzburg-Landau vortices concentrating on curves.

We study the two-dimensional Ginzburg-Landau functional for superconductivity and the related Gross-Pitaevskii functional for Bose-Einstein Condensate. In a convex simply-connected domain, Serfaty has shown that the vortices accumulate around a single point in the domain as the Ginzburg--Landau parameter $\kappa\to\infty$. Our previous papers (with Aftalion and Alama) on multiply connected domains show that vortices may instead accumulate on an appropriate curve as $\kappa\to\infty$. In our recent result with S. Alama and V. Millot, we study the number and distribution of these vortices along the curve of concentration. Their distribution is determined by a classical problem from potential theory.

Posted August 16, 2007

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:40 pm 285 Lockett Hall
Shinnosuke Oharu, Chuo University, Japan

Ecological models of red tide plankton in the coastal ocean.

This talk will be concerned with a mathematical model consisting of an ecological model for a specific species of plankton and an ocean model, numerical models consistent with the PDE models, and computer simulations by means of new CFD methods.

Posted September 14, 2007

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Édouard Oudet, Laboratoire de Mathématiques, Université de Savoie, France

Constant width bodies in dimension 3

A body (that is, a compact connected subset $K$ of $\mathbf{R}^n$) is said to be of *constant width* $\alpha$ if its projection on any straight line is a segment of length $\alpha>0$, the same value for all lines.

We present in this talk a complete analytic parametrization of constant width bodies in dimension 3 based on the median surface: more precisely, we define a bijection between some space of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke's formula. Finally, we present some numerical computations based on the preceding parametrization.

É. Oudet will be visiting the department this week (9/24 – 9/28). If you want to schedule a meeting with him, contact B. Bourdin.

Posted September 28, 2007

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett Hall 233
Burak Aksoylu, Department of Mathematics and CCT

Physics-based preconditioners for solving PDEs on highly heterogeneous media

Eigenvalues of smallest magnitude become a major bottleneck for
iterative solvers especially when the underlying physical properties
have severe contrasts. These contrasts are commonly found in many
applications such as composite materials, geological rock properties
and thermal and electrical conductivity.

The main objective of this work is to construct a method as algebraic
as possible that could efficiently exploit the connectivity of highly
heterogeneous media in the solution of diffusion operators. We
propose an algebraic way of separating binary-like systems according
to a given threshold into high- and low-conductivity regimes of
coefficient size $O(m)$ and $O(1)$, respectively where $m >> 1$. The
condition number of the linear system depends both on the mesh size
$\Delta x$ and the coefficient size $m$. For our purposes, we address
only the $m$ dependence since the condition number of the linear
system is mainly governed by the high-conductivity subblock. Thus, the
proposed strategy is inspired by capturing the relevant physics
governing the problem. Based on the algebraic construction, a
two-stage preconditioning strategy is developed as follows: (1) a
first stage that comprises approximation to the components of the
solution associated to small eigenvalues and, (2) a second stage that
deals with the remaining solution components with a deflation strategy
(if ever needed). The deflation strategies are based on computing near
invariant subspaces corresponding to smallest and deflating them
by the use of recycled the Krylov subspaces.

Due to its algebraic nature, the proposed approach can support a wide
range of realistic geometries (e.g., layered and channelized
media). Numerical examples show that the proposed class of
physics-based preconditioners are more effective and robust compared
to a class of Krylov-based deflation methods on highly heterogeneous
media. We also report on singular perturbation analysis of the stiffness
matrix and the impact of the number of high-conductive regions
on various matrices.

Posted October 9, 2007

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Michael Zabarankin, Stevens Institute of Technology

Generalized Analytic Functions in 3D Axially Symmetric Stokes Flows

A class of generalized analytic functions, defined by a special case of the Carleman system that arises from related potentials encountered in various areas of applied mathematics has been considered. Hilbert formulas, establishing relationships between the real and imaginary parts of a generalized analytic function from this class, have been derived for the domains exterior to the contour of spindle, lens, bi-spheres and torus in the meridional cross-section plane. In bi-spherical and toroidal coordinates, this special case of the Carleman system has been reduced to a second-order difference equation with respect to either the coefficients in series or densities in integral representations of the real and imaginary parts. For spindle and lens, the equation has been solved in the framework of Riemann boundary-value problems in the class of meromorphic functions. For torus, the equation has been solved by means of the Fourier transform, while for bi-spheres, it has been solved by an algebraic method. As examples, analytical expressions for the pressure in the problems of the 3D axially symmetric Stokes flows about rigid spindle, biconvex lens, bi-spheres and torus have been derived based on the corresponding Hilbert formulas.

Posted October 4, 2007

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Thirupathi Gudi, CCT, LSU

Local Discontinuous Galerkin Methods for Elliptic Problems

Posted October 17, 2007

Last modified November 27, 2007

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 233 Lockett
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

Global minimizers for a p-Ginzburg-Landau energy.

We study the problem of existence of global minimizers for a p-Ginzburg-Landau type energy on the plane and on the half-plane, for p>2, under a degree condition at infinity. We prove existence of a minimizer when the degree equals 1. This is joint work with Yaniv Almog, Leonid Berlyand and Dmitry Golovaty.

Posted December 28, 2007

Last modified March 3, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:40 pm 233 Lockett Hall
Peter Sternberg, Indiana University

Bifurcating solutions in a model for a superconducting wire subjected to an applied current

We study formally and rigorously the bifurcation to steady and time-periodic states in a model for a thin superconducting wire in the presence of an imposed current. Exploiting the PT-symmetry of the equations at both the linearized and nonlinear levels, and taking advantage of the collision of real eigenvalues leading to complex spectrum, we obtain explicit asymptotic formulas for the stationary solutions, for the amplitude and period of the bifurcating periodic solutions and for the location of their zeros or “phase slip centers” as they are known in the physics literature. In so doing, we construct a center manifold for the flow and give a complete description of the associated finite-dimensional dynamics. This is joint work with Jacob Rubinstein and Kevin Zumbrun.

Posted February 26, 2008

Last modified March 3, 2008

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett Hall 233
Razvan Teodorescu, Los Alamos National Laboratory

Harmonic Growth in 2D via Biorthogonal Polynomials

Evolution of planar domains (representing physical clusters) under harmonic forces is representative for many problems in mathematical physics. In certain situations, the evolution leads to finite-time singularities. I will discuss a regularization of this evolution inspired by the equilibrium distribution of eigenvalues of large random normal matrices. Connections to operator theory will also be discussed.

Posted March 24, 2008

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 233 Lockett Hall
Yuliya Gorb, Department of Mathematics Texas A&M University

Fictitious Fluid Approach for Justification of Asymptotics of Effective Properties of Highly Concentrated Suspensions

The method of the discrete network approximation has been used for determining effective properties of high contrast disordered composites with particles close to touching. It is illustrated by considering a highly packed suspension of rigid particles in a Newtonian fluid. The effective viscous dissipation rate of such a suspension exhibits a singular behavior, and the goal is to derive and justify its asymptotic formula as a characteristic interparticle distance tends to zero. The main idea of the presented approach is a reduction of the original continuum problem described by partial differential equations with rough coefficients to a discrete network. This reduction is done in two steps which constitute the \"fictitious fluid\" approach. While previously developed techniques based on a direct discretization allowed to obtain only the leading singular term of asymptotics for special symmetric boundary conditions, we are able to capture all singular terms in the asymptotic formula of the dissipation rate for generic boundary conditions. The fictitious fluid approach also allows for a complete qualitative description of microflow in a thin gap between neighboring particles in the suspension.

Posted February 25, 2008

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 237 Lockett Hall
Mikhail Stepanov, Department of Mathematics, The University of Arizona

Instantons in hydrodynamics

We consider the hydrodynamic type system (Navier–Stokes or Burgers equation) with random forcing. The untypical events of high vorticity or large velocity gradients are due to extreme realizations of the forcing. To generate such an event one can increase the forcing amplitude or to optimize its shape (without sacrificing the probability of such forcing to happen). The tails of the velocity field probability distribution function can be obtained by finding an optimal shape of forcing, which corresponds to saddle point (instanton) approximation in the path integral describing the velocity statistics. It will be shown how to find the instantons in hydrodynamic systems numerically.

Posted March 31, 2008

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 240 Lockett Hall
John W. Cain, Virginia Commonwealth University

A Kinematic Model for Propagation of Cardiac Action Potentials

Propagation of cardiac action potentials is usually modeled with a reaction-diffusion equation known as the cable equation. However, when studying the initiation of arrhythmias, one is primarily interested in the progress of action potential wavefronts without regard to the complete wave profile. In this talk, I will explain how to derive a purely kinematic model of action potential propagation in cardiac tissue. I will reduce a standard PDE model (the cable equation) to an infinite sequence of ODEs which govern the progress of wave fronts in a repeatedly stimulated fiber of cardiac tissue. The linearization of the sequence of ODEs admits an exact solution, expressible in terms of generalized Laguerre polynomials. Analyzing the solutions yields valuable insight regarding nonlinear wave propagation in an excitable medium, providing interesting physiological implications.

Posted March 26, 2008

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 233 Lockett Hall
Bogdan Vernescu, Worcester Polytechnic Institute

TBA

Posted September 9, 2008

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Monday, October 27, 2008 TBA
Dmitry Golovaty, University of Akron

TBA

Posted September 13, 2008

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 285
Dmitry Golovaty, University of Akron

An effective model for ferronematic liquid crystals

I will discuss a nonlinear homogenization problem for
ferronematics—colloidal suspensions of small ferromagnetic particles
in a nematic liquid crystalline medium—in a regime when the volume
fraction of weakly interacting particles is small. The energy of the
suspension is given by a Ginzburg–Landau term supplemented by a
Rapini–Papoular surface anchoring energy term and terms describing
interaction between the suspension and the magnetic field. For a pure
nematic, the energy density of interaction between the magnetic field
and the nematic director is given by a quadratic term that is
minimized when the director is parallel to the field. For a
ferronematic, the additional indirect coupling between the nematic
and the field is introduced into the energy via anchoring of nematic
molecules on the surfaces of the particles.

Assuming that the particles are identical prolate spheroids with fixed
positions but variable orientations, we use the method of
quasisolutions to show that the influence of particles on the
suspension can be accounted for by an effective nonlinear potential.
For needle-like particles of large eccentricity, the model reduces to
a known expression of Brochard and de Gennes. This is a joint work
with C. Calderer, A. DeSimone, and A. Panchenko.

Posted October 12, 2008

Last modified October 27, 2008

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 285
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On a minimization problem involving a potential vanishing on two curves

This talk is concerned with a vector-valued singular perturbation problem involving a potential vanishing on two curves. We study the limiting behaviour of the minimizers, and demonstrate how it depends on the geometry of the domain. This is a joint work with Nelly Andre (University of Tours).

Posted November 3, 2008

Last modified November 7, 2008

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 235 Lockett Hall
Phuc Nguyen, Department of Mathematics, Louisiana State University

Quasilinear and Hessian equations with super-critical exponents and singular data

Posted November 19, 2008

Last modified December 5, 2008

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 285
Stephen Shipman, Mathematics Department, LSU

Field sensitivity to L^p perturbations of a scatterer

This will be an informal presentation as part of the weekly material science discussion group. I will discuss the title problem and some related problems I would like to solve.

Posted February 11, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm
Monica Torres, Department of Mathematics, Purdue University

The structure of solutions of systems of hyperbolic conservation laws

Hyperbolic systems of conservations laws model many areas of physics, including fluid mechanics, acoustics, etc. One of the main challenges in the analysis of these equations is that solutions develop singularities even if the initial data is smooth. These singularities are known as shock waves. Existence theorems only show that entropy solutions belong to some $L^p$ space and satisfy an entropy inequality in the distributional sense. Therefore, an open problem is to study the structure of solutions and regularity of the shock waves. In this talk we present results in this direction, which include some Liouville-type results for systems of conservation laws.

Posted January 20, 2009

Last modified April 17, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett Hall 285
David Dobson, Department of Mathematics, University of Utah

Electromagnetic transmission resonances in periodic hole arrays

Recently there has been increasing interest in terahertz-frequency electromagnetic radiation in the engineering community. Improved methods of generating such radiation has led to hopes of applications in communications, imaging, and spectroscopy. Unfortunately almost all materials are highly absorptive in the terahertz range, making device design difficult. One method of manipulating terahertz radiation is by filtering through thin, perforated metal plates. Such plates exhibit interesting, and sometimes unexpected transmission properties. The transmission spectrum depends strongly on both the hole pattern and the aperture shape. This talk will describe some work on developing a model for transmission through periodic hole arrays, including analysis and numerical methods. We conclude with some preliminary work on the problem of optimal design of aperture shape to produce a desired transmission spectrum.

Host: Stephen Shipman

Posted March 13, 2009

Last modified March 15, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm
Tadele Mengesha, Louisiana State University

TBA

Posted April 13, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 285
Tadele Mengesha, Louisiana State University

Sufficient conditions for smooth strong local minima

The talk addresses the conjecture that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a smooth extremal to be a strong local minimizer. Our result holds for a class of variational integrals with integrands of polynomial growth at infinity. The proof is based on the decomposition of an arbitrary variation into its purely strong and weak parts. We show that these two parts act independently on the functional. The action of the weak part can be described in terms of the second variation. While the uniform positivity of the second variation prevents the weak part from decreasing the functional, the uniform quasiconvexity conditions prevent the strong part from doing the same. This is a joint work with Yury Grabovsky.

Posted August 24, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Robert Lipton, Mathematics Department, LSU

Strength of Elastic - Plastic Composites Made From Random Configurations of Two Materials

Posted September 6, 2009

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Room 233 Lockett Hall
Yaniv Almog, Department of Mathematics, LSU

Superconductivity Near the Normal State in the presence of electric current.

We consider the linearization of the time-dependent Ginzburg-Landau near the normal state. We assume that an electric current is applied through the sample, which captures the whole plane, inducing thereby, a magnetic field. We show that independently of the current, the normal state is always stable. Using Fourier analysis the detailed behaviour of solutions is obtained as well. Relying on semi-group theory we then obtain the spectral properties of the steady-state elliptic operator. We shall also consider the spectral properties of the same elliptic operator near a flat wall, and obtain the critical current in the limit of small and large normal conductivity

Posted September 14, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Scott Armstrong, Department of Mathematics, Louisiana State University

Self-similar solution and long-time asymptotics for fully nonlinear parabolic equations

I will present results on the existence and uniqueness of a self-similar solution of a fully nonlinear, parabolic equation (an example of which include the Bellman-Isaacs equation arising in the theory of stochastic optimal control and stochastic differential game theory). As an application, we are able to describe the long-time behavior of solutions to the Cauchy problem, and derive a conservation law which generalizes the conservation of mass in the case of the heat equation. The scaling invariance property of the self-similar solution depends on the nonlinear operator, and is in general different from that of the heat kernel. We will see that this difference has an interesting interpretation in terms of controlled diffusion processes. This work is joint with M. Trokhimtchouk.

Posted August 27, 2009

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 233, Lockett Hall
Matthew Knepley, Computation Institute, University of Chicago

Tree-based methods on GPUs

We examine the performance of the Fast Multipole Method on heterogeneous computing devices, such as a CPU attached to an Nvidia Tesla 1060C card. The inherent bottleneck imposed by the tree structure is ameliorated by a refactoring of the algorithm which exposes the fine-grained dependency structure. Also, common reduction operations are refactored in order to maximize throughput. These optimizations are enabled by our concise yet powerful interface for tree-structured algorithms. Examples of performance are shown for problems arising from vortex methods for fluids

Posted September 14, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Christo Christov, University of Louisiana at Lafayette

Stochastic Functional Expansions for Heterogeneous Continuous Media and Chaotic Regimes of Nonlinear Dynamical Systems

Posted September 20, 2009

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Truyen Nguyen, University of Akron

Hamilton–Jacobi equation in the space of measures associated with a system of conservation laws

We introduce a class of action functional defined over the set of continuous paths in the Wasserstein space of probability measures on $\mathbf{R}^d$. We show that minimizing path for such action exists and satisfies compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton–Jacobi equations are well-posed and their unique viscosity solutions are given by the dynamic programming principle. The characteristics of these Hamilton–Jacobi equations in the space of probability measures are solutions of the compressible Euler equation in $\mathbf{R}^d$. This is joint work with Jin Feng of the University of Kansas.

Posted September 8, 2009

Last modified February 5, 2021

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Rachael Neilan, Department of Oceanography and Coastal Sciences, LSU

Optimal control in disease modeling

Optimal control theory in disease models is used to determine cost-effective disease prevention and treatment strategies. When disease dynamics are governed by ordinary differential equations, Pontryagin's Maximum Principle is used to characterize an optimal control (i.e., optimal treatment strategy). However, many disease models use partial differential equations to describe the spread of infection in space and time. No extension of Pontryagin's Maximum Principle exists for systems of PDEs, but similar techniques are employed to derive optimal spatiotemporal control characterizations. In this talk, we will provide theoretical optimal control results for a system of advection-diffusion equations describing the spread of rabies through a raccoon population. Numerical solutions will illustrate the optimal vaccine distribution on homogeneous and heterogeneous spatial domains.

Posted September 30, 2009

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Scott McKinley, Department of Mathematics, Duke University

Anomalous Diffusion of Distinguished Particles in Bead-Spring Networks. (This is a joint Applied Analysis & Probability Seminar)

Due to recent and compelling experimental observations using
passive microrheology there is theoretical interest in anomalous
sub-diffusion — stochastic processes whose long-term mean-squared
displacement satisfies $\mathbf{E}[x^2(t)] \sim t^\nu$ where $\nu \leq 1$. The
canonical example of a sub-diffusive process is fractional Brownian
motion, but for reasons we will discuss, this project focuses on a
touchstone model from polymer kinetic theory — the Rouse chain — and
its natural generalizations.

Our interest is in studying the dynamics of a distinguished particle
in a network of thermally fluctuating beads that interact with each
other through linear springs. Such processes can be expressed as the
sum of a Brownian motion with a large number of Ornstein–Uhlenbeck
processes. We introduce a single parameter which can be tuned to
produce any sub-diffusive exponent $\nu \in (0,1)$ for the generic
sum-of-OU structure and demonstrate the relationship between this
parameter and the geometric structure of the bead-spring connection
network in which the distinguished particle resides. This development
provides a basis to prove a conjecture from the physics community that
the Rouse exponent $\nu = 1/2$ is universal among a wide class of
models.

Posted September 14, 2009

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Fadil Santosa, Director, Institute for Mathematics and its Applications and School of Mathematics, University of Minnesota

The mathematics of progressive lens design

Progressive addition lenses are prescribed to patients who need correction for both far and near visions. A progressive lens needs to have power that gradually changes from the far vision zone, used for example in driving, and the near vision zone, used for example in reading a map. The basics of optics and lens design will be described. In particular, it will be shown that the problem can be reduced to one of surface design. The surface design problem itself is solved by a variational approach, which can be further simplified by linearization, leading to a fourth order elliptic partial differential equations. Analysis of the resulting equations and development of a computational method are described. Numerical results are presented to illustrate the process of lens design.

Posted September 30, 2009

Last modified November 10, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Xiaoliang Wan, Louisiana State University

A note on stochastic elliptic models

In this talk we will look at two strategies that introduce randomness into elliptic models. One is to treat the coefficient as an spatial random process, which results in an stochastic elliptic model widely used in engineering applications; the other one is to define the stochastic integral using Wick product, which can be regarded as a generalization of Ito integral. The statistics given by these two strategies can be dramatically different. I will compare these two strategies using a one-dimensional problem and present a new stochastic elliptic model to makes them more comparable. Numerical methods will also be discussed.

Posted November 17, 2009

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Santiago Fortes, Department of Mathematics, LSU

Electromagnetic wave propagation in Plasmonic Crystals

The possibility of engineering composite materials with unusual electromagnetic properties (a.k.a. metamaterials) has generated much interest lately. Devices such as invisibility cloaks and perfect lenses could, in principle, be constructed using such materials. I will present a method for obtaining convergent power series representations for the fields and associated dispersion relations of electromagnetic waves propagating in a species of metamaterial known as plasmonic crystal. The technology provided by these series lead to a rich scenario in which to explore effective properties in a mathematically rigorous setting. This has allowed us give definite answers regarding the negative index behavior of plasmonic crystals.

Posted January 19, 2010

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Anna Zemlyanova, Department of Mathematics, LSU

Method of Riemann surfaces in modelling of cavitating flow

Cavitation is the formation of a vapor filled area in the liquid which usually appears due to low pressures and high velocities. Riemann surfaces are used in fluid mechanics both for mathematical modeling of the cavity closure and for solution of the resulting mathematical problems. In this talk I will discuss most commonly used cavity closure models and present a detailed solution to the problem of a supercavitating wedge in a jet or under a free surface using Tulin\'s single- or double-spiral-vortex cavity closure model. The solution involves the application of Riemann-Hilbert problems on the elliptic Riemann surface. The numerical results will be presented.

Posted January 21, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Phuc Nguyen, Department of Mathematics, Louisiana State University

Capacitary inequalities and quasilinear Riccati type equations with critical or super-critical growth

We establish explicit criteria of solvability for the quasilinear Riccati type equation $−\Delta_p u = |∇u|^q + ω$ in a bounded $\mathcal{C}^1$ domain $\Omega ⊂ \mathbb{R}^n$, $n ≥ 2$. Here $\Delta_p$, $p > 1$, is the $p$-Laplacian, $q$ is critical $q = p$ or supper critical $q > p$, and the datum $ω$ is a measure. Our existence criteria are given in the form of potential theoretic or geometric (capacitary) estimates that are sharp when $ω$ is compactly supported in the ground domain $\Omega$. A key in our approach to this problem is capacitary inequalities for certain nonlinear singular operators arising from the $p$-Laplacian.

Posted January 25, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233, Lockett Hall
Alexander Barnett, Department of Mathematics, Dartmouth College

Robust and accurate computation of photonic crystal band structure using a new integral equation representation of quasi-periodic fields

Host: Stephen Shipman

Photonic crystals are dielectric structures with periodicity on the scale of
the wavelength of light. They have a rapidly growing range of applications
to signal processing, sensing, negative-index materials, and the exciting
possibility of integrated optical computing. Calculating their “band
structure” (propagating Bloch waves) is an elliptic PDE eigenvalue problem
with (quasi-)periodic boundary conditions on the unit cell, i.e., eigenmodes
on a torus. Since the material is piecewise homogeneous, boundary integral
equations (BIE) are natural for high-accuracy solution.

In such geometries BIEs are usually periodized by replacement of the free
space Greens function kernel by its quasi-periodic cousin. We show why this
approach fails near the (spurious) resonances of the empty torus. We
introduce a new approach which cures this problem: imposing the boundary
conditions on the unit-cell walls using layer potentials, and a finite
number of neighboring images, resulting in a second-kind integral equation
with smooth data. This couples to existing BIE tools (including high-order
quadratures and Fast Multipole acceleration) in a natural way, allowing
accuracies near machine precision. We also discuss inclusions which
intersect the unit cell walls, and how we use a small number of evaluations
to interpolate over the Brillouin zone to spectral accuracy. Joint work with
Leslie Greengard (NYU).

Posted January 18, 2010

Last modified November 29, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Diego Maldonado, Kansas State University

Bilinear pseudo-differential operators: motivations and recent developments

During the 70s, driven by some problems posed by A. Calderón,
R. Coifman and Y. Meyer pioneered a theory of bilinear pseudo-differential
operators. These operators later found further applications in topics of
analysis and PDEs such as compensated compactness, regularity of solutions
to PDEs, boundedness properties of commutators, bilinear singular
integrals, and paraproducts, and pointwise multipliers for functional
spaces.

Departing from the definition of the Fourier transform, in this talk we
will tour the theory of bilinear pseudo-differential operators and some of
its applications to finally arrive at the latest results and some open
problems.

Posted March 21, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Peter Wolenski, LSU Department of Mathematics
Russell B. Long Professor

Impulsive systems

An impulsive system is a dynamical system that may exhibit “jumps” in the state variable. We shall introduce a model of such systems driven by a measure, and discuss solution concepts and recent results. A model of synaptic dynamics will be given as an example, which has been introduced in the neuroscience literature to describe neuronal population activity.

Posted February 27, 2010

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

3:40 pm 233 Lockett Hall
Vladislav Kravchenko, Dept. of Mathematics, CINVESTAV del IPN, Unidad Querétaro

Solution of boundary and eigenvalue problems for second order elliptic operators in the plane using pseudoanalytic function theory

We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D=divpgrad+q in the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method.

Host: Stephen Shipman

Posted March 21, 2010

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
M. Gregory Forest, Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina
Grant Dahlstrom Distinguished Professor of Mathematics & Biomedical Engineering

Dynamic defect morphology and hydrodynamics of sheared nematic polymers in physically confined geometries

Nematic polymers consist of rigid rod or platelet dispersions where the particles are macromolecules, i.e., larger than liquid crystals but still Brownian. Depending on the properties of the rods or platelets, materials are targeted with extreme barrier, electrical, thermal, mechanical, dielectric or energy storage properties. Unlike fiber processing which yields highly uniform alignment of the rod or platelet phase, film and mold filling processes of nematic polymers typically possess dynamic particle orientational morphology even in steady processing conditions, accompanied by unsteady flow. Furthermore, defects are generic. In this talk we present model equations and boundary conditions, and results from numerical simulations for shear cell and driven cavity experiments of nematic polymers. We use novel defect detection and tracking diagnostics to show defect spawning mechanisms and morphology and flow evolution in these two types of experiments, and sensitivity to boundary conditions as well as initial data. Finally, we report some progress on post-processing of the simulation data to infer the underlying mechanisms for various property enhancements due to the particle phase. This is joint work with several collaborators who will be acknowledged during the lecture.

Posted September 3, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Robert Lipton, Mathematics Department, LSU

Multi-scale analysis and optimal local basis functions for Generalized Finite Element Methods

Modern structures such as airplane wings exhibit complicated sub structures and make use of composite materials in their construction. The high cost of experimental tests for these hierarchical structures is driving a trend toward virtual testing. This requires the development of multi-scale numerical methods capable of handling large degrees of freedom spread across different length scales. In this talk we review multi-scale numerical methods and introduce the theory of the Kolmogorov n-width as a means to identify optimal local basis functions for use in multi-scale finite element methods. We are able to identify a spectral basis with nearly exponential convergence with respect to the dimension of the approximation space. The convergence result is shown to hold in a very general setting. This is joint work with Ivo Babuska.

Posted September 3, 2010

Last modified September 20, 2010

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

Reservoir stimulation: an approach based on variational fracture.

The topic of this talk is to present a first step towards the predictive understanding of the mechanisms used in the creation of the highly connected crack networks required for Enhanced Geothermal Systems and oil shale mining. I will focus on thermal stimulation, where thermal stresses induced by a cold fluid circulating through a hot reservoir lead to nucleation of many short cracks. I will consider the limiting cases of purely diffusive and purely advective heat transfer, corresponding to extreme porosity limits in the reservoir. I will present a mechanistically faithful yet mathematically sound model, based on Francfort and Marigo's generalization of Griffith's idea of competition between bulk and surface energies. I will discuss the virtues of the model, its approximation, and its numerical implementation. Finally, I will present some numerical experiments in 2 and 3 dimensions.

Posted September 10, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Hongchao Zhang, Louisiana State University

A Derivative-free Regularized Trust Region Approach for Least-squares Minimization

We will introduce a class of derivative-free algorithms for the nonlinear least-squares minimization problem. These algorithms are based on polynomial interpolation models and are designed to take advantages of the problem structure. Global and local quadratic convergence properties of the algorithms will be addressed. Promising numerical results compared with other state-of-art software packages indicate the algorithm is very efficient and robust for finding both low and high accuracy solutions.

Posted September 10, 2010

Last modified October 13, 2010

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Tadele Mengesha, Louisiana State University

Weighted and regularity estimates for nonlinear PDEs over rough domains

Global weighted Lp estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. As an application, Morrey and Holder regularity of solutions are established. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation, i.e can have mild discontinuity. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach is a perturbation argument that uses maximal function estimates, Vitali covering lemma, and known regularity results of solutions to nonlinear homogeneous equations. This is a joint work with Nguyen Cong Phuc.

Posted October 23, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 223
Shawn Walker, LSU

Shape Optimization of Chiral Propellers in 3-D Stokes Flow

Locomotion at the micro-scale is important in biology and in industrial applications such as targeted drug delivery and micro-fluidics. We present results on the optimal shape of a rigid body locomoting in 3-D Stokes flow. The actuation consists of applying a fixed moment and constraining the body to only move along the moment axis; this models the effect of an external magnetic torque on an object made of magnetically susceptible material. The shape of the object is parametrized by a 3-D centerline with a given cross-sectional shape. No a priori assumption is made on the centerline. We show there exists a minimizer to the infinite dimensional optimization problem in a suitable infinite class of admissible shapes. We develop a variational (constrained) descent method which is well-posed for the continuous and discrete versions of the problem. Sensitivities of the cost and constraints are computed variationally via shape differential calculus. Computations are accomplished by a boundary integral method to solve the Stokes equations, and a finite element method to obtain descent directions for the optimization algorithm. We show examples of locomotor shapes with and without different fixed payload/cargo shapes.

Posted October 10, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Michael Neilan, Louisiana State University

A unified approach to construct and analyze finite element methods for the Monge–Ampère equation

The Monge–Ampère equation is a fully nonlinear second order PDE that arises in various application areas such as differential geometry, meteorology, reflector design, economics, and optimal transport. Despite its prevalence in many applications, numerical methods for the Monge–Ampère equation are still in its infancy. In this talk, I will first discuss the inherent difficulty of approximating this equation and briefly review the numerical literature. I will then discuss a new approach to construct and analyze several finite element methods for the Monge–Ampère equation. As a first step, I will show that a key feature in developing convergent discretizations is to construct schemes with stable linearizations. I will then describe a methodology for constructing finite elements that inherits this trait and provide two examples: $C^0$ finite element methods and discontinuous Galerkin methods. I will briefly show how to prove the well-posedness of such methods as well as derive optimal order error estimates.

Posted October 25, 2010

Last modified January 6, 2021

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 223
Benjamin Jaye, University of Missouri–Columbia

Quasilinear operators with natural growth terms

Posted October 10, 2010

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Young-Ju Lee, Department of Mathematics Rutgers, The State University of New Jersey

Self-Sustaining Oscillations of the Falling Sphere Through the Johnson-Segalman Fluids

In this talk, we review a novel numerical method that can handle the rate-type non-Newtonian equations in a unified fashion and validate the methods in terms of various benchmark solutions as well as theoretical results. We then apply it to the real physical problems. In particular, we present our investigations and attempts to identify a mathematical model for the unusual phenomenon observed in motion of the sphere falling through the wormlike micellar fluids by Jayaraman and Belmonte; a sphere falling in a wormlike micellar fluids undergoes non-transient and continual oscillations. We tackle the Johnson-Segalman models in the parameter regimes that have been unexplored previously for the flow past a sphere and reproduce the self-sustaining, continual, (ir)regular and periodic oscillations. Our results show that the flow instability can be correlated with the critical value of the velocity gradient, as observed in experiments by Jayaraman and Belmonte in 2003. If time permits, we also present recent works on the boundary conditions for the diffusive complex fluids models as well as the fast stokes solvers implemented in a full parallel fashion.

Posted October 27, 2010

Applied Analysis Seminar Questions or comments?

4:30 pm Lockett 233
Dmitry Golovaty, University of Akron

Coarse-graining in Atomistic Models of Dislocations

Dislocations and their dynamics play a major role in the response of materials to mechanical and thermal loading. Extensive work has been done on diﬀerent scales of the problem from atomistic level, to dislocation level to macroscopic level. Yet the behavior of material under plastic deformation is still a source of many challenging mathematical problems. In this talk we focus on models at the atomistic level and deal with questions of coarse graining

where higher level models are sought. We focus on distribution functions characterizing the atomic arrangement and discuss energy representation and dislocation motion in terms of these statistical properties. The evolution is

formulated as a gradient ﬂow.

Posted November 19, 2010

Last modified February 5, 2021

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Andrés León Baldelli, Université Pierre et Marie Curie

Variational Approach to Fracture Mechanics: Multifissuration and Delamination of Thin Films

Thin film materials such as multilayered composite materials, coating films etc plays a key role in modern engineering applications. The different physical characteristics of the various layers, the production and assembly procedures and the tensile stresses that develop in such systems may induce deformations that can lead to damage and failure. Variational energetic approaches to fracture mechanics [1] has been proved to give a reliable and physically consistent description of these complex phenomena, accurately predicting the experimental results. An extension of this approach to thin film/substrate systems is presented for a film under thermal loads, accounting for the possibility for the film to undergo multifissuration and debonding processes. Analytic results are obtained for the 1D case and compared to those obtained by a FEM approximation.

[1] B. Bourdin, G.A. Francfort, and J.-J. Marigo, "The Variational Approach to Fracture", Springer, 2007.

Posted April 15, 2011

Last modified April 25, 2011

Applied Analysis Seminar Questions or comments?

3:40 pm Lockett 233
Aleksandra Gruszka, LSU Department of Mathematics
PhD Student of Prof. Malisoff

On tracking for the PVTOL model with bounded feedbacks

We study a class of feedback tracking problems for the planar vertical takeoff and landing (PVTOL) aircraft dynamics, which is a benchmark model in aerospace engineering. After a survey of the literature on the model, we construct new feedback stabilizers for the PVTOL tracking dynamics. The novelty of our contribution is in the boundedness of our feedback controllers and their applicability to cases where the velocity measurements may not be available, coupled with the uniform global asymptotic stability and uniform local exponential stability of the closed loop tracking dynamics, the generality of our class of trackable reference trajectories, and the input-to-state stable performance of the closed loop tracking dynamics with respect to actuator errors. Our proofs are based on a new bounded backstepping result. We illustrate our work in a tracking problem along a circle.

Posted April 29, 2011

Applied Analysis Seminar Questions or comments?

3:40 pm Lockett 277
Michael Malisoff, LSU
Roy P. Daniels Professor

Uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high-frequency gains

We study adaptive tracking problems for nonlinear systems with unknown control gains. We construct controllers that yield uniform global asymptotic stability for the error dynamics, and hence tracking and parameter estimation for the original systems. Our result is based on a new explicit, global, strict Lyapunov function construction. We illustrate our work using a brushless DC motor turning a mechanical load. We quantify the effects of time-varying uncertainties on the motor electric parameters.

Note: This talk will be understandable to faculty, staff, students, and visitors who are familiar with the material in Math 7320 (Ordinary Differential Equations) at LSU. No background in control is needed.

Posted April 29, 2011

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett Hall
Cody Pond, Department of Mathematics Tulane University

Effective boundary conditions on insulated bodies

The temperature of perfectly insulated body can be modeled by the heat equation with Neumann (or no-flux) boundary condition. In reality there are no perfect insulators and the actual boundary condition on the body may be only approximately Neumann. In this talk we will see how properties of a layer of insulation affect the boundary condition experienced by the insulated body. We we also see how ignoring physical restrictions in the model can produce some exotic boundary conditions.

Posted August 31, 2011

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm Lockett 233
Catalin Turc, Case Western Reserve University

Fast, high-order solvers based on regularized integral equations for acoustic and electromagnetic scattering problems

We present a class of solvers based on Nystrom discretizations to produce fast and very accurate solutions of acoustic and electromagnetic scattering problems in small numbers of Krylov-subspace iterations. At the heart of our approach is a general methodology that uses certain regularizing operators to deliver integral equation formulations that possess excellent spectral properties for scattering problems, including smooth and non-smooth geometries and a variety of boundary conditions. Our computational methodology relies on a novel Nystrom approach based on use of a overlapping/non-overlapping-patch technique, Chebyshev discretizations and an acceleration method based on equivalent sources and 3D FFT's.

Posted September 30, 2011

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Hanna Terletska, Department of Physics and Astronomy, LSU and Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory

Towards a multiscale formalism for disordered systems

Posted August 23, 2011

Last modified October 23, 2011

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Rustum Choksi, Department of Mathematics and Statistics, McGill University, Montréal, Canada

Self-assembly of Diblock Copolymers and Variational Problems with Long-Range Interactions

Energy-driven pattern formation induced by competing short and long-range interactions is common in many physical systems. This talk will address mathematical and physical paradigms for periodic pattern formation induced by these energetic competitions. The mathematical paradigm consists of nonlocal perturbations to the well-studied Cahn-Hilliard and isoperimetric problems. The physical paradigm is self-assembly of diblock copolymers. Via a combination of analysis and numerics, I will address the structure of minimizers across the phase diagram.

Posted September 12, 2011

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm 233 Lockett
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On the distance between homotopy classes of maps taking values in manifolds

It is well known that for $p ≥ m$ the degree of maps in $W^{1,p}(S^m, S^m)$ is well defined and one has the following decomposition of this space as a disjoint union of homotopy classes: $W^{1,p}(S^m, S^m) = \bigcup_{d\in\mathbb{Z}}\mathcal{E}_d$. It is natural then to study the
distance $δ_p(d_1, d_2)$ between each pair of distinct homotopy classes $\mathcal{E}_{d_1}$ and $\mathcal{E}_{d_2}$, defined by
\[
δ_p^p(d_1, d_2) = \inf\bigl\{ \int_{S^m} |∇(u_1 − u_2)|^p : u_1 \in \mathcal{E}_{d_1},\ u_2\in \mathcal{E}_{d_2} \bigr\}.
\]
In the one dimensional case, $m = 1$, we find that the distance is given explicitly by the formula $δ_p(d_1, d_2) = \tfrac{2^{1+1/p}\,|d_2−d_1|}{π^{1−1/p}}$.

In higher dimensions, $m ≥ 2$, it turns out that in the limiting case $p = m$,
the distance between the homotopy classes is always zero. On the other hand,
when $p > m$, for $d_1 \ne d_2$ the distance is positive, but independent of $d_1$ and $d_2$, i.e., $δ_p(d_1, d_2) = c(m, p)$. Here $c(m, p)$ is a positive constant that had already been computed explicitly by Talenti (for $m = 2$) and Cianchi (for any $m$) in the context of Sobolev-type inequalities on spheres.

This talk is based on a work in progress with Shay Levy and on a earlier work
with Jacob Rubinstein.

Posted August 29, 2011

Last modified February 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 233 Lockett
Leonid V. Berlyand, Department of Mathematics, Pennsylvania State University

Modeling of collective swimming of bacteria.

Bacteria are the most abundant organisms on Earth and they significantly influence carbon cycling and sequestration, decomposition of biomass, and transformation of contaminants in the environment. This motivates our study of the basic principles of bacterial behavior and its control. We have conducted analytical, numerical and experimental studies of suspensions of swimming bacteria. In particular, our studies reveal that active swimming of bacteria drastically alters the material properties of the suspension: the experiments with bacterial suspensions confined in thin films indicate a 7-fold reduction of the effective viscosity and a 10-fold increase of the effective diffusivity of the oxygen and other constituents of the suspending fluid. The principal mechanism behind these unique macroscopic properties is self-organization of the bacteria at the microscopic
level–a multiscale phenomenon. Understanding the mechanism of self-organization in general is a fundamental issue in the study of biological and inanimate system. Our work in this area includes

• *Numerical modeling*. Bacteria are modeled as self-propelled point force dipoles subject to two types of forces: hydrodynamic interactions with the surrounding fluid and excluded volume interactions with other bacteria modeled by a Lennard-Jones-type potential. This model, allowing for numerical simulations of a large number of particles, is implemented on the Graphical Processing Units (GPU), and is in agreement with experiments.

• *Analytical study of dilute suspensions*. We introduced a model for swimming bacteria and obtained explicit asymptotic formula for the effective viscosity in terms of known physical parameters. This formula is compared with that derived in our PDE model for a dilute suspension of prolate spheroids driven by a stochastic torque, which models random reorientation of bacteria (“tumbling”). It is shown that the steady-state probability distributions of single particle configurations are identical for the dilute and semi-dilute models in the limiting case of particles becoming spheres. Thus, a deterministic system incorporating pairwise hydrodynamic interactions and excluded volume constraints behaves as a system with a random stochastic torque. This phenomenon of stochasticity arising from a deterministic system is referred to as self-induced noise.

• *Kinetic collisional model—work in progress*. We seek to capture a phase transition in the bacterial suspension–an appearance of correlations and local preferential alignment with an increase of concentration. Collisions of the bacteria, ignored in most of the previous works, play an important role in this study.

Collaborators: PSU students S. Ryan and B. Haines, and DOE scientists I. Aronson and D. Karpeev (both Argonne Nat. Lab)

Posted November 30, 2011

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Michel Jabbour, University of Kentucky

On step dynamics and related morphological instabilities during epitaxial growth of thin crystalline films.

Thin crystalline films are often bounded by surfaces consisting of flat terraces separated by atomic steps. During epitaxial growth in the step-flow regime, adsorbed atoms (from a vapor or beams) diffuse on the terraces until they attach to steps, causing them to advance. For a train of steps, two modes of morphological instability can occur: bunching, which leads to regions of high step density separated by wide terraces, and meandering, whereby steps become wavy. Experiments indicate that bunching and meandering can coexist on some stepped surfaces, in contrast to the predictions of the standard Burton–Cabrera–Frank (BCF) model. In this talk, I will review the BCF theory and present a thermodynamically consistent (TC) generalization of it that resolves this apparent paradox. In particular, I will show that step bunching and meandering can occur simultaneously, provided that the adatom equilibrium coverage exceeds a critical value. I will also compare the TC model with various extensions of the BCF paradigm that attempt to reconcile theory with experiments.

Posted February 10, 2012

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett, 233
Michael Borden, Institute for Computational Engineering and Sciences, University of Texas at Austin

Isogeometric Analysis and Computational Fracture Mechanics

I will begin my presentation with an overview of isogeometric analysis, emphasizing its application to problems in nonlinear solid mechanics. The basic idea of the isogeometric concept is to use the same basis for analysis as is used to describe the geometry in, for example, a CAD representation. The smoothness of typical geometric representations (e.g., NURBS and T-splines) has been shown to have computational advantages over standard finite elements in many solid mechanics problems.

In the second part of my presentation I will discuss our recent work on the numerical implementation of variational, or phase-field, models of fracture. The phase-field approach to predicting fracture uses a scalar-valued field to indicate that the material is in some state between complete undamaged or completed fractured with a smooth transition between the two states. This allows cracks to be modeled without explicit tracking of discontinuities in the geometry or displacement fields. In this part of my presentation I will also discuss work in which we make use of the smoothness provided by isogeometric analysis to explore the effect of adding higher-order terms to the phase-field model. Several numerical examples will be shown for both two and three-dimensional problems that demonstrate the ability of these models to capture complex crack behavior.

Posted February 5, 2012

Last modified April 30, 2012

Applied Analysis Seminar Questions or comments?

4:10 pm Lockett 233
Shari Moskow, Mathematics Department, Drexel University

Scattering and Resonances of Thin High Contrast Dielectrics

We study the scattered field from a thin high contrast dielectric volume of finite extent. We examine both the Helmholtz model and the full three dimensional time-harmonic equations. For the case of the Helmholtz model, we derive an asymptotic expansion and show error estimates. We also consider the problem of calculating resonance frequencies by using these asymptotics and compare it with using finite elements and perfectly matched layers. For Maxwell equations, we derive a formulation of Lippmann-Schwinger type which has an additional surface term to account for the discontinuities. We analyze this surface term and present the limiting equations that result.

(based on joint work with collaborators D. Ambrose, J. Gopalakrishnan, F. Santosa and J. Zhang)

Posted March 5, 2012

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Lockett 233
Richard Lehoucq, Sandia National Laboratories

A new approach for a nonlocal, nonlinear conservation law

My presentation describes an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators introduced do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we examine a nonlocal inviscid Burgers equation, which gives a basic form with which to characterize well-posedness. We describe the connection to a nonlocal viscous regularization, which mimics the viscous Burgers equation in an appropriate limit. We present numerical results that compare the behavior of the nonlocal Burgers formulation to the standard local case. The developments presented in this paper form the preliminary building blocks upon which to build a theory of nonlocal advection phenomena consistent within the peridynamic theory of continuum mechanics. This is joint work with Qiang Du (PSU), Jim Kamm (SNL) and Mike Parks (SNL)

Posted August 8, 2012

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm 233 Lockett
Raul Tempone, King Abdullah University of Science and Technology, KAUST

Strategies for Optimal Polynomial Approximation of Elliptic PDEs with Stochastic Coefficients

Partial differential equations with stochastic coefficients are a suitable tool to describe systems whose parameters are not completely determined, either because of measurement errors or intrinsic lack of knowledge on the system. In the case of elliptic PDEs, an effective strategy to approximate the state variables and their statistical moments is to introduce high order polynomial approximations like Stochastic Galerkin or Stochastic Collocation method, exploiting the fact that the state variables may exhibit high regularity in their dependence with respect to the random parameters. When the number of parameters is moderate, these methods can be remarkably more effective than classical sampling methods. However, contrary to the latter, the performance of polynomial approximations deteriorates as the number of random variables increases (\\emph{curse of dimensionality}); to prevent this, care has to be put in the construction of the approximating polynomial space. In this talk we will propose strategies to construct optimal spaces and propose some particular polynomial spaces and generalized sparse grids that are optimal for particular problems. We will also support our claims with some simple numerical examples. This work is a joint collaboration with J. Beck, F. Nobile and L. Tamellini.

Posted August 7, 2012

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

4:40 pm - 5:30 pm 233 Lockett Hall
Mohammad Motamed, Visiting Scholar, Institute for Engineering and Computational Science UT Austin

Analysis and Computation of Linear Hyperbolic Problems with Random Coefficients

In this talk, in particular, we consider the second-order acoustic and elastic wave equations. In the first part of this talk, we propose and analyze a stochastic collocation method for solving the acoustic wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We demonstrate different types of convergence of the “probability error” with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In the second part of the talk, we present extensions to the elastic wave equation with random coefficients and random boundary conditions.

Posted September 4, 2012

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

4:00 pm Lockett 233
Stephanos Venakides, Department of Mathematics, Duke University

Higher breaking in the focusing nonlinear Schrödinger equation

The focusing nonlinear Schrödinger equation iε du/dt + ε^{2} d^{2}u/dx^{2} + 2|u|^{2}u = 0 (NLS) appears dominantly in nonlinear optical transmission, together with its many variants. Mathematically, the initial value problem of the NLS on the line is integrable. It can be linearized with the aid of a Lax operator pair, produced by Zakharov and Shabat. Determining the evolution of an NLS waveform becomes possible with the aid of Riemann-Hilbert problems (RHP), the conceptual nature of which is simple and will be explained in the talk.

The development of the steepest descent method for oscillatory RHP provided rigorous asymptotic procedures, that make the solution of NLS and nonlinear integrable systems in general, explicit or nearly explicit. The method applies to asymptotics of RHP in the same spirit as the classic methods of stationary phase and steepest descent apply in the asymptotic evaluation of Fourier type integrals arising from the solution of linear differential equations. In both the linear and the nonlinear cases, there is a separation of space-time scales over similar parameter regimes.

Employing initial data of the form u(x,0)=A(x)exp(iS(x)/ε) in the asymptotic limit ε→0, we describe the solution over a large domain of space-time and the mechanism of the break-down of the method beyond this domain. Using a combination of analytic and numerical considerations, we establish the boundary beyond which the asymptotic solution is still unknown. The spatial component of this domain is bounded.

Posted October 7, 2012

Last modified May 5, 2020

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett
Stewart Silling, Sandia National Laboratories

Multiscale Modeling of Fracture with Peridynamics

The peridynamic theory is an extension of traditional solid mechanics that treats discontinuous media, including the evolution of discontinuities such as fracture, on the same mathematical basis as classically smooth media. Since it is a strongly nonlocal theory, peridynamic material models contain a length scale that characterizes the interaction distance between material points. By changing this length scale in a way that preserves the bulk elastic properties, greater spatial resolution in a simulation can be focused on a growing crack tip or other evolving singularity. This leads to a consistent way to treat fracture at the smallest physically relevant length scale within a larger model, without remeshing or coupling dissimilar methods. The method works within a meshless discretization of the peridynamic equations similar to that used in the Emu code. The grid has multiple elves of resolution. The high resolution portions of the grid supply material properties, including damage, to the coarser levels. The displacement field in the coarsest level is determined by the equation of the motion at that level, using these coarse-grained material properties. The resulting coarse displacements are applied as boundary conditions on the finer levels of the grid. The equation of the motion in the finer levels is solved only where the damage is ongoing or large deformations are occurring. In this way, the greatest computational power is focused only on those parts of the region, such as growing crack tips, where it is required. This peridynamic multiscale method appears to provide a promising approach to understanding the evolution of material failure, including the interaction of small defects with each other and with heterogeneities. This talk will first review the basics of the peridynamic theory. The new multi scale method will then be discussed, with computational examples drawn from the mechanics of contact and from damage progression in heterogeneous media.

Posted September 19, 2012

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:00 pm Lockett 233
Anna Zemlyanova, Department of Mathematics, Texas A&M

Elimination of oscillating singularities at the crack-tips of an interface crack with a help of a curvature-dependent surface tension

A new model of fracture mechanics incorporating a curvature-dependent surface tension acting on the boundaries of a crack is considered. The model is studied on the example of a single straight interface crack between two elastically dissimilar semi-planes. Linear elasticity is assumed for the behavior of the material of the plate in a bulk. A non-linear boundary condition with a consideration for a curvature-dependent surface tension is given on the crack boundary. It is well known from linear elastic fracture mechanics (LEFM) that oscillating singularities exist at the crack tips and lead to non-physical interpenetration and wrinkling of the crack boundaries. Using the methods of complex analysis, such as Dirichlet-to-Neumann mappings, the problem is reduced to a system of six singular integro-differential equations. It is proved that the introduction of the curvature-dependent surface tension eliminates both classical power singularities of the order 1/2 at the tips of the crack and oscillating singularities, thus resolving the classical contradictions of LEFM. Numerical computations are presented.

Posted October 29, 2012

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:40 pm - 4:30 pm Room 233 Lockett
Michael Malisoff, LSU
Roy P. Daniels Professor

Asymptotic Stabilization for Feedforward Systems with Delayed Feedbacks

We study a problem of state feedback stabilization of time-varying feedforward systems with a pointwise delay in the input. Our approach relies on a time-varying change of coordinates and Lyapunov-Krasovskii functionals. Our result applies for any given constant delay, and provides uniformly globally asymptotically stabilizing controllers of arbitrarily small amplitude. The closed-loop systems enjoy input-to-state stability properties with respect to additive uncertainty on the controllers. We illustrate our work using a tracking problem for a model for high level formation flight of unmanned air vehicles. We will review all of the necessary background on control theory, so no prior exposure to controls will be needed to understand this talk. This work is joint with Frederic Mazenc from INRIA in France.

Posted November 26, 2012

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Timothy Healey, Cornell University
Professor of Mathematics and Mechanical and Aerospace Engineering

Nonlinear problems for thin elastic structures and the ubiquitous isola bifurcation

We begin with a simple 1-dimensional, 2-phase solid under "hard" tensile end loading in the presence of inter-facial effects. This is equivalent to a phase-field model, of the van der Waals-Cahn-Hilliard type, that illustrates well the concept of an isola bifurcation. Roughly speaking, the latter corresponds to the nucleation, growth, decay and eventual disappearance of a stable, inhomogeneous solution (representing here a phase mixture) as the loading parameter is monotonically increased. We then present results for three ostensibly distinct problems (models) - all exhibiting this same isola-bifurcation phenomenon: (i) twining in shape-memory solids; (ii) two-phase configuration of GUV's (fluid-elastic shell models of lipid-bilayer vesicles); (iii) wrinkling of highly stretched, finely thin rectangular sheets.

Posted March 8, 2013

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233 Originally scheduled for Monday, March 11, 2013
Stefan Llewellyn Smith, University of California, San Diego

Hollow Vortices

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. After giving a history of point vortices, we obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

Posted February 23, 2013

Last modified April 9, 2013

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Daniel Onofrei, University of Houston

Active control of acoustic and electromagnetic fields

The problem of controlling acoustic or electromagnetic fields is at the core of many important applications such as, energy focusing, shielding and cloaking or the design of supper-directive antennas. The current state of the art in this field suggests the existence of two main approaches for such problems: passive controls, where one uses suitable material designs to control the fields (e.g., material coatings of certain regions of interest), and active control techniques, where one employs active sources (antennas) to manipulate the fields in regions of interest. In this talk I will first briefly describe the main mathematical question and its applications and then focus on the active control technique for the scalar Helmholtz equation in a homogeneous environment. The problem can be understood from two points of view, as a control question or as an inverse source problem (ISP). This type of ISP questions are severely ill posed and I will describe our results about the existence of a unique minimal energy solution. Stability of the solution and extensions of the results to the case of nonhomogeneous environment and to the Maxwell system are part of current work and will be described accordingly.

Posted May 8, 2013

Last modified October 1, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Guillermo Ferreyra, Mathematics Department, LSU

The Future of Analysis at LSU

SCI Data [PDF]

Letter from Provost [DOCX]

Posted October 15, 2013

Last modified October 17, 2013

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 233
Stephen Shipman, Mathematics Department, LSU

Efficient Evaluation of 2D-periodic Green functions in 3D

I will describe the analytical basis behind a fast method of computing periodic Green functions, ultimately for the purpose of efficiently solving problems of scattering by periodic structures. The Poisson summation formula provides super-algebraic convergence away from frequencies for which one of the Rayleigh-Bloch modes is grazing. At grazing (cutoff) frequencies, the periodic Green function ceases to exist, and a more complicated method is needed. This involves introducing several sheets of periodic sources to create a half-space Green function. This is work with Oscar Bruno, Catalin Turc, and Stephanos Venakides.

Posted October 24, 2013

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Room 233 Lockett Hall
Robert Lipton, Mathematics Department, LSU

Dynamics in Materials Far From Equilibrium

In this talk we address the role of local instability in the precipitation and propagation of failure in macroscopic samples of material. We work with non-locally interacting systems, eg. peridynamics. A class of scaled nonlinear interaction potentials are identified for which dynamic instability localizes and fracture surfaces appear in the scaling limit.

Posted October 25, 2013

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett Hall 233
Yaniv Almog, Department of Mathematics, LSU

Global stability of the normal state of superconductors under the effect of strong electric current

Consider a superconducting wire whose temperature is lower than the critical one. When one flows a sufficiently strong current through the wire, it is well known from experimental observation that the wire becomes resistive, behaving like a normal metal. We prove that the time-dependent Ginzburg-Landau model anticipates this behaviour. We first prove that, for sufficiently strong currents, the semi-group associated with the model, becomes a contraction semi-group. Then, we obtain an upper bound for the critical current where the semi-group becomes stable. We relate this current to the resolvent of the linearized elliptic operator. Joint work with Bernard Helffer

Posted October 24, 2013

Applied Analysis Seminar Questions or comments?

3:00 pm - 4:00 pm Lockett Hall 233
Amit Acharya, Carnegie Melon University

PDE dynamics of dislocations

The talk will describe a PDE framework to deal with the dynamics of dislocations leading to plasticity in solids. Dislocations are defects of deformation compatibility/integrability in elastic response. The presented framework will be shown to be capable of representing discrete defect dynamics as well as present a natural setting for asking questions related to macroscopic plasticity arising from the underlying dislocation dynamics.

Posted October 24, 2013

Applied Analysis Seminar Questions or comments?

3:00 pm - 4:00 pm Lockett Hall 233
Amit Acharya, Carnegie Melon University

PDE dynamics of dislocations

The talk will describe a PDE framework to deal with the dynamics of dislocations leading to plasticity in solids. Dislocations are defects of deformation compatibility/integrability in elastic response. The presented framework will be shown to be capable of representing discrete defect dynamics as well as present a natural setting for asking questions related to macroscopic plasticity arising from the underlying dislocation dynamics.

Posted February 5, 2014

Last modified January 24, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

Asymptotics of eigenstates of elliptic problems with mixed boundary data in domains tending to infinity

We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show, in particular, that in general the limiting behavior is very different from the one with Dirichlet boundary conditions. This is a joint work with Michel Chipot and Prosenjit Roy from the University of Zurich.

Posted February 19, 2014

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett Hall Room 233
Michael Mascagni, Department of Computer Science, Florida State University

Random Number Generation Using Normal Numbers

Normal numbers are irrationals with perfect digit distribution, and thus they are potentially sources of computational random numbers. Among provably normal numbers are the Stoneham numbers, which are special not only in their digit distribution, but in the fact that finite segments of the digits can be quickly computed. Thus, we examine random numbers produced by periodic sections of the digits of Stoneham numbers. We show how they are equivalent to a linear congruential generator with special parameters, and we investigate this generator as a linear congruential generator. This is joint work with Steve F. Brailsford.

Posted February 5, 2014

Last modified February 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
Bernard Helffer, University of Paris South, Orsay

Introduction to spectral minimal partitions, Aharonov-Bohm's operators and Pleijel's theorem

Given a bounded open set $Ω$ in $\mathbb{R}^n$ (or in a Riemannian manifold) and a partition $\mathcal{D}$ of $Ω$ by $k$ open sets $D_j$ , we can consider the quantity $Λ(\mathcal{D}) := \text{max}_j λ(D_j)$ where $λ(D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $\mathfrak{L}_k(Ω)$ the infimum over all the $k$-partitions of $Λ(\mathcal{D})$ a minimal $k$-partition is then a partition which realizes the infimum. Although the analysis is rather standard when $k = 2$ (we find the nodal domains of a second eigenfunction), the analysis of higher $k$’s becomes non trivial and quite interesting.

Posted April 3, 2014

Last modified April 21, 2014

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 233 Lockett Hall
Yuri Antipov, Mathematics Department, LSU

Diffraction of an obliquely incident electromagnetic wave by an impedance wedge

Posted March 26, 2014

Last modified April 21, 2014

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Mark Wilde, LSU Department of Physics/CCT

Renyi generalizations of the conditional quantum mutual information

Abstract: The conditional quantum mutual information I(A;B|C) of a tripartite quantum state on systems ABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A;B|C)=I(A;B|D) for a four-party pure state on systems ABCD. It has been an open question to find Renyi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different Renyi generalizations of the conditional mutual information that all converge to the conditional mutual information in a limit. Furthermore, we prove that many of these generalizations satisfy the aforementioned properties. As such, the quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Renyi conditional mutual informations defined here with respect to the Renyi parameter. We prove that this conjecture is true in some special cases and when the Renyi parameter is in a neighborhood of one. Finally, we discuss how our approach for conditional mutual information can be extended to give Renyi generalizations of an arbitrary linear combination of von Neumann entropies, particular examples including the multipartite information and the topological entanglement entropy. This is joint work with Mario Berta (Caltech) and Kaushik Seshadreesan (LSU). This is based on the recent paper http://arxiv.org/abs/1403.6102

Posted August 21, 2014

Last modified February 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
Phuc Nguyen, Department of Mathematics, Louisiana State University

The Navier-Stokes equations in nonendpoint borderline Lorentz spaces

It is shown both locally and globally that $L_t^∞(L_x^{3,q})$ solutions
to the three-dimensional Navier-Stokes equations are regular
provided $q\ne ∞$. Here $L_x^{3,q}$, $0 < q \le ∞$, is an increasing
scale of Lorentz spaces containing $L_x^3$. Thus the result provides an
improvement of a result by Escauriaza, Seregin and Šverák ((Russian) Uspekhi Mat. Nauk **58** (2003), 3–44; translation in Russian Math. Surveys **58** (2003), 211–250), which treated the case $q = 3$. A new local energy bound
and a new $\epsilon$-regularity criterion are combined with the
backward uniqueness theory of parabolic equations to obtain the
result. A weak-strong uniqueness of Leray-Hopf weak solutions
in $L_t^∞(L_x^{3,q})$, $q\ne ∞$, is also obtained as a consequence.

Posted September 8, 2014

Last modified September 25, 2014

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Cristi Guevara, LSU Department of Mathematics

Characterization of finite-energy solutions to the focusing 2-dimensional quintic NLS equation

Abstract. In this lecture we will focus on the mass-supercritical and energy-subcritical nonlinear Schroedinger equation or 2 dimensional quintic NLS. Using the concentration-compactness and rigidity method developed by Kenig-Merle, we characterize global behavior of solutions with H1 (finite energy) initial data. In particular, we will discuss an application of the concentration-compactness to the existence of weak blowup solutions for infinite-variance initial data. In addition, we will describe extensions on the conditions for scattering of globally existing solutions for the d-dimensional case.

Posted October 22, 2014

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

4:30 pm Lockett 223
Martin Adler, University of Tübingen

Perturbations of generators of C_0-semigroups

The theory of strongly continuous semigroups is an elegant method to investigate the well-posedness of abstract Cauchy problems. After introducing the basic theory of C_0-semigroups needed for this approach, I provide an overview of bounded and unbounded perturbation results. Finally, we will apply this theory to a delay equation.

Posted November 1, 2014

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Room 233 Lockett
Jacob Grey, Department of Mathematics LSU

A qualitative analysis of some Nonlinear Dispersive Evolution Equations

Posted November 12, 2014

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Room 233 Lockett
Michael Malisoff, LSU
Roy P. Daniels Professor

Designs and Theory for State-Constrained Nonlinear Feedback Controls for Delay Systems: An Infomercial

This talk will discuss some of my research that is being supported by my two new research grants from the US National Science Foundation Directorate for Engineering. The first grant project is entitled “Robustness of Networked Model Predictive Control Satisfying Critical Timing Constraints” and focuses on resolving contentions in a class of communication networks that are common in automobiles and other real-time control applications, and is joint with the Georgia Institute of Technology School of Electrical and Computer Engineering. The second project, “Designs and Theory of State-Constrained Nonlinear Feedback Controls for Delay and Partial Differential Equation Systems,” covers control designs for classes of ordinary and hyperbolic partial differential equations that arise in oil production and rehabilitation engineering, and is joint with the University of California, San Diego Department of Mechanical and Aerospace Engineering. In the first 10 minutes, I will provide a brief description of the basic ideas of control theory. Then, I will present a 25 minute summary of my research on neuromuscular electrical stimulation (or NMES), which is a biomedical approach for helping to restore movement in patients with mobility disorders. My NMES research designed controls for NMES of the human knee under delays and subject to a constraint on the allowable knee position, and is joint with my PhD student Ruzhou Yang and with Prof. Marcio de Queiroz, who are with the LSU Department of Mechanical and Industrial Engineering. In the last 10 minutes, I will advertise for my open PhD student positions on my grants, by providing a brief nontechnical summary of the problems to be addressed and discussing the role PhD students would play in the research. This talk will be accessible to students and others who are familiar with basic differential equations. No background in controls is needed.

Posted February 23, 2015

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

Asymptotic behavior of critical points of an energy involving a "circular-well" potential

We study the singular limit of critical points of an energy with a penalization term depending on a small parameter. The energy involves a potential which is a nonnegative function on the plane, vanishing on a closed curve. We generalize to this setting results obtained by Bethuel, Brezis and Helein for the Ginzburg-Landau energy. This is a joint work with Petru Mironescu (Lyon I).

Posted February 15, 2015

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett Hall Room 233
Robert P. Viator, Jr., LSU

Perturbation Theory of High-Contrast Photonic Crystals

Posted February 15, 2015

Last modified March 8, 2015

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Room 233 Lockett Hall
Anthony Polizzi, LSU

An asymptotic formula for solutions to a heterogeneous logistic equation with small diffusion rate

Of concern is a semilinear elliptic boundary value problem on a bounded domain with a smooth boundary. Its solution u(x) represents the steady state population density of a species in an insulated habitat with a given resource profile a(x) and a given nonnegative coefficient representing the species' rate of random diffusion. We study asymptotic expansions for solutions in the form of Taylor series in the diffusion coefficient. It turns out that, in the presence of diffusion, u(x) depends analytically on the diffusion coefficient, which trivializes the convergence of such a series. We therefore focus our attention primarily on the more delicate case of zero diffusion, in which u(x) is not analytic in the coefficient. It is known that u(x) tends to a(x) as the coefficient tends to zero. We generalize this result by rigorously establishing the desired expansions under suitable assumptions on a(x). Our main result is their convergence on the closure of the domain in this case. We also give an explicit formula for each coefficient of the expansions.

Posted February 13, 2015

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm 233 Lockett Hall
Aleksandr Smirnov, Department of Mathematics, LSU

A discrete model of a fracture in an inhomogeneous strip

Posted February 3, 2015

Applied Analysis Seminar Questions or comments?

3:30 am - 4:30 pm 233 Lockett Hall
Yuri Antipov, Mathematics Department, LSU

Singular integral equations in a segment with two fixed singularities and applications

Posted April 29, 2015

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Room 233 Lockett Hall
Kaushik Dayal, Carnegie Mellon University

A Dynamic Phase-field Model for Structural Transformations and Twinning: Regularized Interfaces with Transparent Prescription of Complex Kinetics and Nucleation

Phase-field models enable easy computations of microstructure because they
regularize sharp interfaces. In addition, the nucleation of new interfaces
and the kinetics of existing interfaces occurs “automatically” using only
the energy and a gradient descent dynamics. This automatic nucleation and
kinetics is often cited as an advantage of these models, and is not present
in sharp interface approaches where nucleation and kinetics must be
explicitly prescribed.

However, this is not necessarily an advantage. Rather, it does not allow
us to use nucleation and kinetic insights that may be gained from
experiment and/or molecular simulations. Hence, this feature is actually a
disadvantage because it breaks the multiscale modeling hierarchy of feeding
information through the scales. Motivated by this, we have developed a
phase-field model (i.e., with regularized interfaces) that allows for easy
and transparent prescription of kinetics and nucleation. We present the
formulation of the model, and characterization through various examples.

Posted May 3, 2015

Applied Analysis Seminar Questions or comments?

3:30 am Lockett 233
Rainer Nagel, University of Tübingen

Some Operator Theoretic Aspects of Ergodic Theory

Abstract: We discuss some techniques and results on linear operators in Banach spaces as, e.g., appearing in the proof of the Green-Tao Theorem on arithmetic progressions in the primes. The main object is the so called Koopman operator yielding a linear model of a nonlinear dynamical system. It is joint work with Tanja Eisner, Balint Farkas and Markus Haase appearing as Springer Graduate Text in Mathematics.

Posted October 4, 2015

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 223
Bálint Farkas, University of Wuppertal

The periodic decomposition problem for semigroups

Given commuting power-bounded linear operators T_{1},...,T_{n} on a Banach space the periodic decomposition problems, originally due to I.Z. Ruzsa, asks whether and under which conditions the equality ker (T_{1}-I) ··· (T_{n}-I) = ker(T_{1}-I)+···+ker (T_{n}-I) holds true. In this talk we focus also on the case when T_{j}=T(t_{j}), t_{j} >0, j=1,..., n for some (strongly continuous) one-parameter semigroup (T(t))_{t≥0}. Moreover, we look at a generalization of the periodic decomposition problem when instead of the cyclic semigroups {T_{j}^{n}:n ∈ **N**} more general semigroups of bounded linear operators are considered.

Posted October 9, 2015

Last modified October 13, 2015

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 232
Ko-Shin Chen, U. Conn.

Ginzburg-Landau and Gross-Pitaevskii Vortices on Surfaces

We consider the Ginzburg-Landau energy on compact and simply-connected surfaces. The first result is the instability of critical points of the Ginzburg-Landau energy. We show on a surface without boundary, any non-constant critical points must be unstable for small epsilon if at least one limiting vortex is located at a point of positive Gauss curvature. The second is the vortex dynamics for the Ginzburg-Landau heat flow, both in the asymptotic regime where the parameter 'epsilon' attends to zero and for a fixed epsilon. We show the vortices of a solution evolve according to the gradient flow of the renormalized energy. Then we establish vortex annihilation results for both ODE and PDE settings. The third is a similar analysis of vortex motion for the Gross-Pitaevskii equation. We show the vortices of a solution follow the Hamiltonian point-vortex flow associated with the renormalized energy. Then on surfaces of revolution, we find rotating periodic solutions to the generalized point-vortex problem and seek a rotating solution to the Gross-Pitaevskii equation having vortices that follow those of the point-vortex flow for small epsilon.

Posted October 26, 2015

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 223
Boris Baeumer, University of Otago, New Zealand

Anomalous Reaction-Diffusion Equations

Abstract: We show how a simple random walk model can be build up step by step to lead to a Volterra integral equation problem whose kernel depends on its solution. The build-up includes fractional differential equations, continuous time random walk limits, and surprising reaction effects. Variants or special cases of the model have been used to describe phenomena in cell dynamics, ecology, epidemiology, and hydrology.

Posted March 21, 2016

Last modified April 5, 2016

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Gianni Royer Carfagni, Università degli Studi Di Parma, Department of Civil Engineering, Environmental Engineering, and Architecture

Phase-field description of structured deformations in plasticity

Abstract: A variational approach to determine the deformation of an ideally plastic substance is proposed by solving a sequence of energy minimization problems under proper conditions to account for the irreversible character of plasticity. The flow is driven by the local transformation of elastic strain energy into plastic work on slip surfaces, once that a certain energetic barrier for slip activation has been overcome. The distinction of the elastic strain energy into spherical and deviatoric parts can also be used to incorporate in the model the idea of von Mises plasticity and isochoric plastic strain. This is a "phase field mode" because the matching condition at the slip interfaces are substituted by the evolution of an auxiliary phase field that, similarly to damage theory, is unitary on the elastic phase and null on the yielded phase. The slip lines diffuse in bands whose width depends upon a material length-scale parameter. Numerical experiments on representative problems in plane strain give solutions with striking similarities with the results from classical slip-line field theory of plasticity, but the proposed model is much richer because, accounting for elastic deformations, it can describe the formation of slip bands at the local level, which can nucleate, propagate, widen and diffuse by varying the boundary conditions.

Posted May 2, 2016

Applied Analysis Seminar Questions or comments?

1:30 pm - 2:30 pm Lockett 233
Kim Pham, ENSTA ParisTech

Construction of a macroscopic model of phase-transformation for the modeling of superelastic Shape Memory Alloys

Abstract: Shape Memory Alloys (SMA) e.g. NiTi display a superelastic behavior at high temperature. Initially in a stable austenite phase, SMA can transform into an oriented martensite phase under an applied mechanical loading. After an unloading, the material recovers its initial stress-free state with no residual strain. Such loading cycle leads to an hysteresis loop in the stress-strain diagram that highlights the dissipated energy for having transformed the material. In a rate-independent context, we first show how a material stability criterion allows to construct a local one-dimensional phase transformation model. Such models relies on a unique scalar internal variable related to the martensite volume fraction. Evolution problem at the structural scale is then formulated in a variational way by means of two physical principles: a stability criterion based on the local minima of the total energy and an energy balance condition. We show how such framework allows to handle softening behavior and its compatibility with a regularization based on gradient of the internal variable. We then extend such model to a more general three dimensional case by introducing a tensorial internal variable. We derive the evolution laws from the stability criterion and energy balance condition. Second order conditions are presented. Illustrations of the features of such model are shown on different examples.

Posted September 23, 2016

Last modified March 3, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Viktoria Kuehner, University of Tübingen

Semiflows and Koopman semigroups

We characterize Koopman semigroups $(T(t))_{t\geq 0}$ on $\mathrm{L}^1(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a standard probability space, induced by a measurable semiflow $(\varphi_t)_{t\geq 0}$ on $X$, by means of their generator $(A,D(A))$. We then construct a topological model $(\psi_t)_{t\geq 0}$ of that semiflow on a compact space $K$ such that the Koopman semigroup induced by the continuous semiflow $(\psi_t)_{t\geq 0}$ is isomorphic to the original semigroup.

Posted December 2, 2016

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Malcolm Brown, Department of Computer Science & Informatics, Cardiff University

Scattering and inverse scattering for a left-definite Sturm–Liouville problem

This talk reports on recent work which develops a scattering and an inverse scattering theory for the Sturm–Liouville equation $u'' + qu = \lambda w u$, where $w$ may change sign but $q$ is positive. Thus the left-hand side of the equation gives rise to a positive quadratic form, and one is led to a left-definite spectral problem. The crucial ingredient of the approach is a generalised transform built on the Jost solutions of the problem and hence termed the “Jost transform” and the associated Paley–Wiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the Camassa–Holm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for $u'' + qu = \lambda w u$.

This is joint work with Christer Bennewitz (Lund, Sweden) and Rudi Weikard (Birmingham, AL).

Posted September 28, 2016

Last modified February 1, 2017

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
David Shirokoff, New Jersey Institute of Technology

Approximate global minimizers to pairwise interaction problems through a convex/non-convex energy decomposition: with applications to self-assembly

Abstract: A wide range of particle systems are modeled through energetically driven interactions, governed by an underlying non-convex and often non-local energy. Although numerically finding and verifying local minima to these energies is relatively straight-forward, the computation and verification of global minimizers is much more difficult. Here computing the global minimum is important as it characterizes the most likely self-assembled arrangement of particles (in the presence of low thermal noise) and plays a role in computing the material phase diagram. In this talk I will examine a general class of model functionals: those arising in non-local pairwise interaction problems. I will present a new approach for computing approximate global minimizers based on a convex/non-convex splitting of the energy functional that arises from a convex relaxation. The approach provides a sufficient condition for global minimizers that may in some cases be used to show that lattices are exact, and also be used to estimate the optimality of any candidate minimizer. Physically, the approach identifies the emergence of new length scales seen in the collective behavior of interacting particles. (This is a joint Applied Analysis/Computational Mathematics seminar.)

Posted September 1, 2017

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Yaniv Almog, Department of Mathematics, LSU

On a Schrödinger operator with a purely imaginary potential in the semiclassical limit

We consider the operator ${\mathcal A}_h=-h^2\Delta+iV$ in the semi-classical limit $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.

Posted August 25, 2017

Last modified September 8, 2017

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
Changfeng Gui, University of Texas at San Antonio

The Sphere Covering Inequality and its applications

In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two {\it distinct} surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least $4 \pi$. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. The talk is based on joint work with Amir Moradifam from UC Riverside.

Posted August 9, 2017

Last modified September 21, 2017

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
Giles Auchmuty, University of Houston

The SVD of the Poisson kernel

The Poisson kernel provides a representation for the solution operator for the Dirichlet problem for Laplace's equation on a bounded region. It is usually treated as an integral operator and this talk will describe spectral representations of this operator when the boundary data is in L^2(\partial\Omega). For this problem Fichera (1955) proved, under strong regularity conditions on the boundary, that the Poisson kernel is a continuous linear transformation of L^2(\partial\Omega) to L^2(\Omega) and that it has norm related to the first eigenvalue of a Steklov eigenproblem for the biharmonic operator on \Omega. In this talk two quite different representations of this operator using Steklov eigenfunctions and Hilbert space theory will be outlined. The first is based on the use of harmonic Steklov eigenfunctions. They may be used to develop a different theory of boundary trace spaces such as H^s(\partial\Omega). This yields spectral representation of solutions of Robin and Neumann boundary value problems for Laplace's equation as well as the Dirichlet problem. There are associated approximation theories and generalizations of results such as the mean value theorem to rectangles and boxes. When the domain is a ball, the results provide an analysis in terms of classical spherical harmonics. A weak version of the Dirichlet Biharmonic Steklov eigenproblem that Fichera studied will be described using Hilbert-Sobolev space methods. It can be shown that the normal derivatives of these eigenfunctions provide an orthonormal basis of L^2(\partial \Omega) while their Laplacians provide an L^2 orthogonal basis of harmonic functions on \Omega. This yields an SVD of the Poisson kernel and the norm of the operator is related to the first Steklov eigenvalue of the problem.

Posted August 7, 2017

Last modified October 7, 2017

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 134
Keng Deng, University of Louisiana at Lafayette

Global existence and blow-up for nonlinear diffusion equations with boundary flux governed by memory

In this talk, we introduce the study of global existence and blow-up in finite time for nonlinear diffusion equations with flux at the boundary governed by memory. Via a simple transformation, the memory term arises out of a corresponding model introduced in previous studies of tumor-induced angiogenesis. The study is also in the spirit of extending work on models of the heat equation with local, nonlocal, and delay nonlinearities present in the boundary flux. Specifically, we establish an identical set of necessary and sufficient conditions for blow-up in finite time as previously established in the case of local flux conditions at the boundary.

Posted August 25, 2017

Last modified October 10, 2017

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 134
Ryan Hynd, University of Pennsylvania

Partial regularity for doubly nonlinear parabolic systems

We will present a regularity result for solutions of a PDE system which is a model for general doubly nonlinear evolutions. The system we focus on a particular case of a general class of flows that arise in the study of phase transitions.

Posted September 13, 2017

Applied Analysis Seminar Questions or comments?

2:30 pm - 3:20 pm Lockett 233
Jiahong Wu, Oklahoma State University

Partial differential equations related to fluids with partial or fractional dissipation

There have been substantial recent developments on several partial differential equations from fluid dynamics with partial or fractional dissipation. This talk summarizes results on the global existence and regularity problem for the 3D Navier-Stokes equations with partial hyperdissipation, the surface quasi-geostrophic equation, the 2D Boussinesq equations with partial or fractional dissipation and the 2D magnetohydrodynamic equations with partial or fractional dissipation.

Posted August 15, 2017

Last modified October 11, 2017

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
Kun Zhao, Tulane University

Analysis of a System of Parabolic Conservation Laws Arising From Chemotaxis

In contrast to random diffusion without orientation, chemotaxis is the biased movement of organisms toward the region that contains higher concentration of beneficial or lower concentration of unfavorable chemicals. The former often refers to the attractive chemotaxis and latter to the repulsive chemotaxis. Chemotaxis has been advocated as a leading mechanism to account for the morphogenesis and self-organization of a variety of biological coherent structures such as aggregates, fruiting bodies, clusters, spirals, spots, rings, labyrinthine patterns and stripes, which have been observed in many laboratory experiments. Mathematical modeling of chemotaxis was initiated more than half a century ago. The Keller-Segel type model has provided a corner for much of the works investigating chemotaxis, its success being its intuitive simplicity, analytical tractability and capability of modeling the basic phenomena in chemotactic populations. In this talk, I will present a group of results concerning the rigorous analysis of a system of parabolic conservation laws derived from a Keller-Segel type chemotaxis model with singular sensitivity. In particular, global well-posedness, long-time asymptotic behavior, zero chemical diffusion limit and boundary layer formation of classical solutions will be discussed.

Posted August 15, 2017

Last modified January 24, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 233
Zhifu Xie, University of Southern Mississippi

Variational method with SPBC and Broucke-Hénon orbit and Schubart orbit

N-body problem concerns the motion of celestial bodies under universal gravitational attraction. Although it has been a long history to apply variational method to N-body problem, it is relatively new to make some important progress in the study of periodic solutions. We develop the Variational Method with Structural Prescribed Boundary Conditions (SPBC) and we apply it to study periodic solutions in the 3-body problem with equal masses. We show that under an appropriate topological constraint, the action minimizer must be either the Schubart orbit (1956) or the Broucke-Hénon orbit (1975). One of the main challenges is to prove that the Schubart orbit coincides with the action minimizer connecting a collinear configuration with a binary collision and an isosceles configuration which must be collinear. A geometric property of the action minimizer is introduced to overcome this challenge. The action minimizer without collisions can be extended to the Broucke-Hénon orbit.

Posted November 30, 2017

Last modified December 3, 2017

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Michael Malisoff, LSU
Roy P. Daniels Professor

Stability And Robustness Analysis For A Multispecies Chemostat Model With Delays

Abstract: The chemostat is a laboratory device and a mathematical model for the continuous culture of microorganisms. Chemostat models have been studied extensively, because of their importance in biotechnology and ecology. This talk will discuss a chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allow delays in the growth rates and additive uncertainties. Using constant inputs of certain species as controls, we derive bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. Under delays and uncertainties, we provide bounds on the delays and on the uncertainties that ensure input-to-state stability with respect to uncertainties. No prerequisite background in biology or control theory will be necessary to understand and appreciate this talk.

Posted January 11, 2018

Last modified February 20, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Wei Li, LSU

Fluorescence ultrasound modulated optical tomography in diffusive regime

Fluorescence optical tomography (FOT) is an imaging technology that localizes fluorescent targets in tissues. FOT is unstable and of poor resolution in highly scattering media, where the propagation of multiply-scattered light is governed by the smoothing diffusion equation. We study a hybrid imaging modality called fluorescent ultrasound-modulated optical tomography (fUMOT), which combines FOT with acoustic modulation to produce high-resolution images of optical properties in the diffusive regime. The principle of fUMOT is to perform multiple measurements of photon currents at the boundary as the optical properties undergo a series of perturbations by acoustic radiation, in which way internal information of the optical field is obtained. We set up a Mathematical model for ufUMOT, prove well-posedness for certain choices of parameters, and present reconstruction algorithms and numerical experiments for the well-posed cases.

Posted January 10, 2018

Last modified February 12, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Masato Kimura, Kanazawa University, Japan

A phase field model for crack propagation and some applications

Posted January 16, 2018

Last modified October 1, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Tadele Mengesha, The University of Tennessee, Knoxville

Sobolev regularity estimates for solutions to spectral fractional elliptic equations

Global Calderón-Zygmund type estimates are obtained for solutions to fractional elliptic problems over smooth domains. Our approach is based on the "extension problem" where the fractional elliptic operator is realized as a Dirichlet-to-Neumann map corresponding to a degenerate elliptic PDE in one more dimension. This allows the possibility of deriving estimates for solutions to the fractional elliptic equations from that of degenerate elliptic equations. We will confirm this first by obtaining weighted estimates for the gradient of solutions to a class of linear degenerate/singular elliptic problems over a bounded, possibly non-smooth, domain. The class consists of those with coefficient matrix that symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a particular weight that belongs to a Muckenhoupt class. The weighted estimates are obtained under a smallness condition on the mean oscillation of the coefficients with a weight. This is a joint work with T. Phan.

Posted January 10, 2018

Last modified February 5, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Prashant Kumar Jha, LSU

Numerical analysis of finite element approximation of nonlocal fracture models

We discuss nonlocal fracture model and present numerical analysis of finite element approximation. The peridynamic potential considered in this work is the regularized version of the bond-based potential generally considered in peridynamic literature (Silling 2000). In the limit of vanishing nonlocality, peridynamic model behaves like a elastodynamic model away from a crack zone and has a finite fracture energy associate to crack set (Lipton 2014, 2016).Using this property we relate the parameters in a peridynamic potential with given elastic constant and fracture toughness. Before we consider finite element approximation, we show that the problem is well posed. We show the existence of evolutions in H^2 space. We consider finite element discretization in space and central difference in time to approximate the problem. Approximation is shown to converge in L^2 norm at the rate Ct\triangle t+C_sh^2/s^2. Here \triangle t is the size of time step, h is the mesh size, and is the size of horizon (nonlocal scale). Constants C_t and C_s are independent of h and \triangle t. In the absence of nonlinearity, stability of approximation is shown. Numerical results are presented to verify the convergence rate. This is a joint work with Robert Lipton.

Posted March 18, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Stephen Shipman, Mathematics Department, LSU

Reducibility of the Fermi surface for periodic quantum-graph operators

The Fermi, or Floquet, surface for a periodic operator at a given energy level is an algebraic variety that describes all complex wave vectors admissible by the periodic operator at that energy. Its reducibility is intimately related to the construction of embedded eigenvalues supported by local defects. The rarity of reducibility is reflected in the fact that a generic polynomial in several variables cannot be factored. The "easy" mechanism for reducibility is symmetry. However, reducibility ensues in much more general and interesting situations. This work constructs a class of non-symmetric periodic Schrodinger operators on metric graphs (quantum graphs) whose Floquet surface is reducible. The graphs in this study are obtained by coupling two identical copies of a periodic quantum graph by edges to form a bilayer graph. Reducibility of the Floquet surface for all energies ensues when the coupling edges have potentials belonging to the same asymmetry class, that is, when their "spectral A-functions" are identical. If the potentials of the connecting edges belong to different asymmetry classes, then typically the Floquet surface is not reducible. Bilayer graphene is a notable exception--its Floquet surface is always reducible.

Posted April 2, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Ivan Gudoshnikov, The University of Texas at Dallas

Stabilization of quasistatic evolution of elastoplastic systems subject to periodic loading

We consider an arrangement of m elastoplastic springs (elastoplastic system) that are connected according to a given graph. Each spring i is described by both elastic e_i and plastic p_i strains, but only the elastic strains e_i generate stress responses s_i. We develop an analytic framework to design time-periodic loadings which make the evolution s(t) of the stress vector s = (s_1, ..., s_m) converging to a globally asymptotically stable time-periodic regime.
The core of our approach is in converting the problem into a sweeping process with a moving polyhedron, which was earlier proposed by Moreau [C.I.M.E. notes, 1974]. We prove that global stability of a unique periodic regime takes place if the moving polyhedron is a simplex, which we further link to a simple topological property of the elastoplastic system under consideration.
To illustrate the abstract theorem, sample sweeping processes are solved numerically by the so-called catch-up algorithm (which we implement using a constrained quadratic optimization pack-age).

The preprint is available at https://arxiv.org/abs/1708.03084.

Posted August 13, 2018

Last modified September 2, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232
Ian Wood, School of Mathematics, University of Kent

Boundary triples and spectral information in abstract M-functions

Abstract: The Weyl-Titchmarsh m-function is an important tool in the study of forward and inverse problems for ODEs, as it contains all the spectral information of the problem. The abstract setting of boundary triples allows the introduction of an abstract operator M-function. It is then interesting to study how much spectral information is still contained in the M-function in this more general setting. Boundary triples allow for the study of PDEs, block operator matrices and many other problems in one framework. We will discuss properties of M-functions, their relation to the resolvent and the spectrum of the associated operator, and connections to the extension theory of operators.

Posted August 29, 2018

Applied Analysis Seminar Questions or comments?

Lockett 232
Malcolm Brown, Department of Computer Science & Informatics, Cardiff University

Spectral problems on star graphs

We report on a two-step reduction method for spectral problems on a star graph with n+1 edges and a self-adjoint matching condition at the central vertex . The first step is a reduction to the problem on a single edge but with an energy depending boundary condition at the vertex. In the second step, by means of an abstract inverse result for m-functions, , a reduction to a problem on a path graph with two edges joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realisations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer the question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type? This is joint work with Heinz Langer and Christine Tretter

Posted September 12, 2018

Last modified September 21, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Aynur Bulut, LSU

Logarithmically energy-supercritical Nonlinear Wave Equations: axial symmetry and global well-posedness

In nonlinear dispersive PDE, radial symmetry often plays a key role in allowing for more refined analysis of the nonlinear interactions which could lead to possible blowup. We will describe recent work where we have recently introduced a mechanism for relaxing assumptions of radiality by considering symmetry in a subset of the variables (for instance, assuming that the initial data is axially symmetric). We applied this philosophy to show global well-posedness and scattering in for the nonlinear wave equation in the logarithmically energy-supercritical setting, generalizing a result of Tao which was established for the radial case. The uses Morawetz and Strichartz estimates that have been adapted to the new symmetry assumption. These methods in fact bring a new perspective to sharp estimates for the energy-critical problem, along the lines of the influential work of Ginibre, Soffer, and Velo. This is joint work with B. Dodson

Posted August 31, 2018

Last modified February 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 232
Karthik Adimurthi, Seoul National University

Partial existence result for Homogeneous Quasilinear parabolic problems beyond the duality pairing

In this talk, we will discuss the existence theory of distributional solutions solving \[ \begin{cases} u_t − \text{div}\,\mathcal{A}(x, t, ∇u) = 0&\text{on } Ω × (0, T),\\ u = u_0&\text{on } ∂Ω × (0, T),\\ u = 0&\text{on } Ω × {t = 0}, \end{cases} \] on a bounded domain $Ω$. The nonlinear structure $\mathcal{A}(x, t, ∇u)$ is modeled after the standard parabolic $p$-Laplace operator. In order to do this, we develop suitable techniques to obtain a priori estimates between the solution and the boundary data. As a consequence of these estimates, a suitable compactness argument can be developed to obtain the existence result. An interesting ingredient in the proof is the careful use of the boundedness of the Hardy-Littlewood Maximal function in negative Sobolev spaces.

Posted August 25, 2018

Last modified August 29, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Jiuyi Zhu, LSU

Nodal sets for Robin and Neumann eigenfunctions

We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the nodal sets is obtained for the Robin eigenfunctions. For the analytic domains, we show a sharp upper bound for the nodal sets on the boundary of the Robin and Neumann eigenfunctions. Furthermore, the sharp doubling inequality and vanishing order are obtained.

Posted September 13, 2018

Last modified October 18, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Blaise Bourdin, Department of Mathematics, Louisiana State University

Variational phase-field models of fracture

Since their inception, over 20 years ago, variational phase-field models of fracture have become widely popular. Part of their success is undoubtedly due to their ability to be efficiently implemented in two and three space dimension, and to their demonstrated ability to capture complex fracture behavior in a wide range of situations. In this presentation, I will go back to the roots of this family of models, deriving Francfort and Marigo's variational approach to fracture from Griffith's classical theory. I will construct variational phase-field models as a numerical approximation for this approach. I will present numerical simulation highlighting the properties of this approximation, as well as some that cannot be fully explained by the mathematical theory. I will then describe an alternate construction as gradient-damage models can explain this behavior and will show how this dual view can address some of the long standing issues in the modeling of brittle solids, including crack nucleation and size effect. Finally, I will discuss ongoing extensions and open issues.

Posted August 29, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Robert Lipton, Mathematics Department, LSU

Predicting complex fracture evolution using nonlocal dynamics

The dynamic fracture of brittle solids is a particularly interesting collective interaction connecting both large and small length scales. Apply enough stress or strain to a sample of brittle material and one eventually snaps bonds at the atomistic scale leading to fracture of the macroscopic specimen. We discuss a nonlocal model for calculating dynamic fracture. The force interaction is derived from a double well strain energy density function, resulting in a non- monotonic material model. The material properties change in response to evolving internal forces eliminating the need for a separate phase field to model the fracture set. (However there is no free lunch and the discrete model is posed in terms of a dense matrix and parallel computation must be used to solve fracture problems.) The model can be viewed as a regularized fracture model. In the limit of zero nonlocal interaction, the model recovers a sharp interface evolution characterized by the classic Griffith free energy of brittle fracture with elastic deformation satisfying the linear elastic wave equation off the crack set. We conclude with a numerical analysis of the model which is joint work with Prashant Jha.

Posted August 29, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Robert Lipton, Mathematics Department, LSU

Understanding nonlocal models for fracture simulation

The peridynamic model is increasingly being used and developed for fracture simulation. In this talk we go "under the hood" to see how nonlocal models can capture the fracture process and to see how they relate to existing fracture models. Along the way we show how the peridynamic energy is related to the Griffiths fracture energy and how the nonlocal evolution satisfies the principle of least action.

Posted September 10, 2018

Last modified October 30, 2018

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Yuri Antipov, Mathematics Department, LSU

Method of automorphic functions for an inverse problem of antiplane elasticity

A nonlinear inverse problem of antiplane elasticity (theory of harmonic functions) for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings for circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky symmetry groups a series-form representation of a ($3n-4$)-parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.

Posted September 12, 2018

Last modified January 24, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Wei Li, LSU

Embedded eigenvalues for the Neumann–Poincaré operator

The Neumann–Poincaré operator and its adjoint are boundary-integral operators associated with harmonic layer potentials. We proved the existence of eigenvalues in the essential spectrum for the Neumann–Poincaré operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann–Poincaré operator for curves of class C^2 with the continuous spectrum generated by a corner. Even (odd) eigenfunctions are proved to lie within the continuous spectrum of the odd (even) component of the operator when a C^2 curve is perturbed by inserting a small corner.

Posted March 11, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Alexander Ioffe, Technion

Topics in Variational Analysis

Posted April 10, 2019

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Peter Wolenski, LSU Department of Mathematics
Russell B. Long Professor

Fully convex Bolza problems with state constraints and impulses

A Fully Convex Bolza (FCB) problem has the appearance of the classical calculus of variations Bolza problem \[ \min \int_0^T L(x(t),x'(t))\,dt + l(x(0),x(T)) \] where the minimization is over $x()$ belonging to some class of arcs. The distinguishing features of FCB are that the data $L(,)$ and $l(,)$ (i) may take on the value infinity and (ii) are convex functions. Allowance of (i) provides great flexibility incorporating constraints so that most standard control problems come under its purlieu. However, broad generality is restrained by (ii), which although quite special, nonetheless includes the classical linear quadratic regulator and many of its generalizations. Furthermore, (ii) opens up the applicability of the tools of convex analysis. We shall review the Hamilton–Jacobi (HJ) theory for FCB problems when the data has no implicit state constraints and is coercive, in which case the minimizing class of arcs are Absolutely Continuous (AC). When a state constraint $x(t) \in X$ is added to the problem formulation, the dual variable may exhibit an impulse or “jump” when the constraint is active. The two properties of a state constraint and noncoercive data (which induce impulsive behavior) are in fact dual to each other, and the minimizing class becomes those of bounded variation. We shall describe Rockafellar's optimality conditions for these problems and a new technique for approximating them by AC problems that utilizes Goebel's self-dual envelope. The approximating AC problems maintain duality and the existing theory can be applied to them. It is proposed that an HJ theory can be developed for BV problems as an appropriate limit of the approximating AC problems. An explicit example will illustrate this.

Posted September 4, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Wei Li, LSU

Fluorescence ultrasound modulated optical tomography (fUMOT) in the radiated transport regime with angularly averaged measurements

We consider an inverse transport problem in fluorescence ultrasound modulated optical tomography (fUMOT) with angularly averaged illuminations and measurements. We study the uniqueness and stability of the reconstruction of the absorption coefficient and the quantum efficiency of the fluorescent probes. Reconstruction algorithms are proposed and numerical validations are performed. This is joint work with Yang Yang and Yimin Zhong, and it is an extension of our previous work done in 2018, where a diffusion model for this problem was considered.

Posted September 13, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Isaac Michael, Louisiana State University

Weighted Birman-Hardy-Rellich type Inequalities with Refinements

In 1961, Birman proved a sequence of inequalities valid for functions in C_0^{n}((0, \infty)) containing the classical (integral) Hardy inequality and the well-known Rellich inequality. Over the years there has been much effort in improving these inequalities with weights and singular logarithmic refinement terms. Using a simple variable transformation in integrals, we prove a generalization of these inequalities involving unrestricted power-type weights and logarithmic refinement terms, on both the exterior interval (R, \infty) and the interior interval (0, R) for any finite R>0. This is based on joint work with Fritz Gesztesy, Lance Littlejohn, and Michael Pang.

Posted September 4, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Phuc Nguyen, Department of Mathematics, Louisiana State University

TBA

Posted October 2, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Phuc Nguyen, Department of Mathematics, Louisiana State University

Weighted and pointwise bounds in measure datum problems with applications

Muckenhoupt-Wheeden type bounds and pointwise bounds by Wolff's potentials are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in the sense of Reifenberg. The principal operator here is modeled after the $p$-Laplacian, where for the first time a singular case is considered. As an application, sharp existence and removable singularity results are obtained for a class of quasilinear Riccati type equations having a gradient source term with linear or super-linear power growth. This talk is based on joint work with Quoc-Hung Nguyen.

Posted November 3, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Andrei Tarfulea, Louisiana State University

The Boltzmann equation with slowly decaying initial data

In this talk we look at the Boltzmann equation, a kinetic continuum model for plasma and high-energy gases. We will look at some previous results on well-posedness for the Cauchy problem, before presenting our recent result on local well-posedness (for a much wider range of parameters) with reduced assumptions on the initial data. As an application, our theorem combines with preexisting results to yield a continuation criterion for the larger parameter range. The scope of the talk will be to examine some of the various difficulties and methodologies associated with the Boltzmann collision operator: the physical symmetries, decompositions, and geometric lemmas needed to control (and in some cases extract regularity from) this nonlinear nonlocal interaction.

Posted October 5, 2019

Last modified October 22, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett Room 233
Matthias Maier, Department of Mathematics Texas A&M University

Simulation of Optical Phenomena on 2D Material Devices

In the terahertz frequency range, the effective (complex-valued) surface conductivity of atomically thick 2D materials such as graphene has a positive imaginary part that is considerably larger than the real part. This feature allows for the propagation of slowly decaying electromagnetic waves, called surface plasmon-polaritons (SPPs), that are confined near the material interface with wavelengths much shorter than the wavelength of the free-space radiation. SPPs are a promising ingredient in the design of novel optical devices, promising "subwavelength optics" beyond the diffraction limit. There is a compelling need for controllable numerical schemes which, placed on firm mathematical grounds, can reliably describe SPPs in a variety of geometries. In this talk we present a number of analytical and computational approaches to simulate SPPs on 2D material interfaces and layered heterostructures. Aspects of the numerical treatment such as absorbing perfectly matched layers, local refinement and a-posteriori error control are discussed. We show analytical results for some prototypical geometries and a homogenization theory for layered heterostructures.

Posted November 8, 2019

Last modified November 9, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
F. Alberto Grünbaum, University of California, Berkeley

Quantum walks: a nice playground for old and new mathematics.

I will give an ab-initio talk trying to show how some time honored pieces of analysis can be used to answer questions about recurrence of quantum walks. I will start with a quick review of classical random walks and then show how Schur functions (the same I. Schur of many other deep topics) are useful in the quantum case. Recently these Schur functions have been seen to be useful in getting a topological classification of quantum walks that respect certain symmetries but go beyond the translation invariant case. I will not assume any previous knowledge about quantum walks. This is joint work with Jean Bourgain, Luis Velazquez, Reinhard Werner, Albert Werner and Jon Wilkening.

Posted September 6, 2019

Last modified November 24, 2019

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Isaac Michael, Louisiana State University

On Weighted Hardy-Type Inequalities

We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman--Hardy--Rellich-type inequalities and derive an operator-valued version thereof. This is based on joint work with C. Chuah, F. Gesztesy, L. L. Littlejohn, T. Mei, M. H. Pang

Posted January 11, 2020

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm 232 Lockett Hall
Quoc-Hung Nguyen, ShanghaiTech University

Quantitative estimates for Lagrangian flows associated to non-Lipschitz vector fields

Since the work by DiPerna and Lions (89) the continuity and transport equation under mild regularity assumptions on the vector field have been extensively studied, becoming a florid research field. In this talk, we give an overview of this theory presenting classical results and new quantitative estimates. One important tool in our investigation is a Kakeya type singular operator. We establish the weak type (1,1) bound for this operator and we exploit it to prove well-posedness and stability results for the continuity and transport equation associated to vector fields represented as singular integrals of BV functions. We also discuss the optimality of this result. Finally, we present sharp regularity estimates for solutions of the continuity equation under various assumptions on the velocity fields.

Posted January 30, 2020

Last modified February 2, 2020

Applied Analysis Seminar Questions or comments?

4:30 pm - 5:20 pm Lockett 233 Originally scheduled for Thursday, January 30, 2020
Khai Nguyen, NCSU

The metric entropy for nonlinear PDEs

Inspired by a question posed by Lax in 2002, in recent years it has received an increasing attention the study on the metric entropy (epsilon entropy) for nonlinear PDEs. In this talk, I will present recent results on sharp estimates in terms of epsilon entropy for hyperbolic conservation laws and Hamilton-Jacobi equations. Estimates of this type play a central role in various ares of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of "resolution" of a numerical method for the corresponding equations.

Posted December 2, 2019

Last modified March 2, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Junshan Lin, Auburn University

Scattering Resonances Through Subwavelength Holes and Their Applications in Imaging and Sensing

The so-called extraordinary optical transmission (EOT) through metallic nanoholes has triggered extensive research in modern plasmonics, due to its significant applications in bio-sensing, imaging, etc. The mechanisms contributing to the EOT phenomenon can be complicated due to the multiscale nature of the underlying structure. In this talk, I will focus on mechanisms induced by scattering resonances. In the first part of the talk, based upon the layer potential technique, asymptotic analysis and the homogenization theory, I will present rigorous mathematical analysis to investigate the scattering resonances for several typical two-dimensional structures, these include Fabry-Perot resonance, Fano resonance, spoof surface plasmon, etc. In the second part of the talk, preliminary mathematical studies for their applications in sensing and super-resolution imaging will be given. I will focus on the resonance frequency sensitivity analysis and how one can achieve super-resolution by using subwavelength structures.

Posted December 2, 2019

Last modified March 8, 2020

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Rudi Weikard, University of Alabama at Birmingham

Topics in inverse problems of differential equations

Rudi Weidard's research interests are currently in Inverse Problems. He also investigates differential equations in the complex domain and in abelian functions.

Posted October 25, 2020

Last modified October 29, 2020

Applied Analysis Seminar Questions or comments?

3:30 pm https://lsu.zoom.us/j/93208607251?pwd=RVRzeE1BSmFnZXEwMEVsdmVicnYxdz09
Khang Huynh, UCLA

A geometric trapping approach to global regularity for 2D Navier-Stokes on manifolds

We use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier-Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our arguments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schrodinger equation, and ideas from microlocal analysis.

This is joint work with Aynur Bulut.

Posted October 28, 2020

Last modified November 29, 2020

Applied Analysis Seminar Questions or comments?

3:30 pm https://lsu.zoom.us/j/92851843655?pwd=dGRGMzIvSmt2UEZwa3g1TmJGVnZTQT09
Edriss Titi, University of Cambridge, Texas A&M University, and Weizmann Institute of Science

Recent Advances Concerning the Navier-Stokes and Euler Equations

In this talk I will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness using the convex integration machinery and their physical relevance. Moreover, I will show the universality of the critical $1/3$ H\"older exponent, conjectured by Onsager for the preservation of energy in Euler equations, by extending the Onsager conjecture for the preservation of generalized entropy in general conservation laws. In addition, I will present a blow-up criterion for the 3D Euler equations based on a class of inviscid regularization for these equations and the effect of physical boundaries on the potential formation of singularity.

Posted January 21, 2021

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (https://lsu.zoom.us/j/91681134143)
Li Chen, LSU

L^p Poincaré inequalities on nested fractals

On nested fractals such as Sierpinski gasket and Vicsek sets, neither analogue of curvature nor differential structure exists. In this setting, I will discuss scale invariant L^p Poincaré inequalities on metric balls in the range 1 ≤ p ≤ 2, using an essentially heat kernel based approach. This is joint work with Fabrice Baudoin.

Posted January 21, 2021

Last modified February 7, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (Link: https://lsu.zoom.us/j/91327987785)
Bjoern Bringmann, UCLA

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity

In this talk, we discuss the construction and invariance of the Gibbs measure for a three- dimensional wave equation with a Hartree-nonlinearity. In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual singularity of the Gibbs measure and the so-called Gaussian free field. In contrast, the Gibbs measure for one or two-dimensional wave equations is absolutely continuous with respect to the Gaussian free field. In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. At the moment, this is the only theorem proving the invariance of any singular Gibbs measure under a dispersive equation.

Posted January 24, 2021

Last modified February 14, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (Link: https://lsu.zoom.us/j/98145526481)
Phillip Isett, UT Austin

A Proof of Onsager's Conjecture for the Incompressible Euler Equations

In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration using the "Mikado flows" introduced by Daneri-Székelyhidi with a new "gluing approximation" technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.

Posted January 24, 2021

Last modified February 15, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom link: https://lsu.zoom.us/j/5494314978
Tuoc Phan, University of Tennessee–Knoxville

Regularity theory in Sobolev spaces for parabolic-elliptic equations with singular degenerate coefficients

We consider a class of second order elliptic and parabolic equations in the upper-half space in which the coefficients can be singular or degenerate on the boundary as of prototype x_d alpha, where alpha is a real number. Two boundary value problems are considered: the zero flux one and the homogeneous Dirichlet one. Corresponding to each of the two problem, generic weighted Sobolev spaces are found to establish the existence, uniqueness, and regularity estimates solutions. As alpha may not be in (-1, 1), our weight x_d alpha may not be in the A_2 Muckenhoupt class of weight as commonly considered in literature. Moreover, the results demonstrate that different weighted Sobolev spaces are required for the two different boundary conditions, a phenomenon that is not seen in the case that the coefficients are uniformly elliptic. The talk is based on joint work with Hongjie Dong (Brown University).

Posted January 24, 2021

Last modified February 22, 2021

Applied Analysis Seminar Questions or comments?

3:30 am Zoom link: https://lsu.zoom.us/j/5494314978
Jinping Zhuge, University of Chicago

Large-scale regularity for stationary Navier-Stokes equations over non-Lipschitz boundaries

We consider the stationary Navier-Stokes equations over a mesoscopically oscillating John boundary (with a non-slip boundary condition), which is non-Lipschitz and allows (inward) cusps or fractals. With such low regularity on the oscillating boundary, we show a large-scale Lipschitz estimate for the velocity and a large-scale oscillation estimate for the pressure. By introducing the 1st-order and 2nd-order boundary layers, we also show large-scale C^{1,alpha} and C^{2,alpha} estimates (For C^{2,alpha} estimate, we assume the boundary is periodic). The proofs rely on the quantitative excess decay method developed recently in homogenization theory. This is joint work with Christophe Prange and Mitsuo Higaki.

Posted January 26, 2021

Last modified March 2, 2021

Applied Analysis Seminar Questions or comments?

3:30 am Zoom link: https://lsu.zoom.us/j/5494314978
Zhongwei Shen, University of Kentucky

Sharp Convergence Rates for Darcy's Law

In this talk I will describe a recent work on the convergence rates for Darcy's law. We consider the Dirichlet problem for the steady Stokes equations in a periodically perforated and bounded domain. We establish the sharp convergence rate for the solutions as the period converges to zero. This is achieved by constructing two boundary correctors to control the boundary layers created by the incompressibility condition and the discrepancy of the boundary values. One of the correctors deals with the tangential boundary data, while the other handles the normal boundary data.

Posted February 3, 2021

Last modified March 22, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (Link: https://lsu.zoom.us/j/91327987785?pwd=VlAxM0QvSDdPY1JSOHlwamVQVWdJQT09)
Yu Deng, USC

Random tensor and nonlinear dispersive equations

We discuss recent developments in the random data theory for nonlinear dispersive equations. In particular, we introduce the methods of random averaging operators and random tensors, which have been used to solve the full 2D Gibbs measure problem, and prove almost-sure well-posedness results at optimal regularity. This is joint work with Andrea R. Nahmod and Haitian Yue.

Posted February 3, 2021

Last modified March 28, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (Link: https://lsu.zoom.us/j/98145526481)
Victor Lie, Purdue University

The LGC-method

Posted February 19, 2021

Last modified April 4, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (Link https://lsu.zoom.us/j/98145526481)
Philippe Sosoe, Cornell University

Optimal integrability threshold for the Gibbs measure associated to the focusing NLS on the torus

In a seminal influential paper, Lebowitz, Rose and Speer (1988) constructed measures on periodic functions inspired by the Gibbs measures of statistical mechanics and based on Brownian motion. These measures are naturally associated to the focusing mass-critical nonlinear Schroedinger equation. They conjectured that these measures are invariant under the nonlinear flow. This was later proved by Bourgain. Lebowitz-Rose-Speer also proposed a critical mass threshold past which the measure no longer exists, given by the mass of the ground state on the real line.

With T. Oh and L. Tolomeo, we prove the optimality of this critical mass threshold. The proof also applies to
the two-dimensional radial problem posed on the unit disc. In this case, this answer a question posed by Bourgain and Bulut (2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is normalizable *at* the optimal mass threshold, answering another posed by Lebowitz, Rose, and Speer (1988). This latter fact is somewhat surprising in view of the minimal mass blowup solution for the focusing quintic nonlinear Schroedinger equation on the one-dimensional torus.

Posted April 9, 2021

Last modified November 5, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (Link: https://lsu.zoom.us/j/91327987785?pwd=VlAxM0QvSDdPY1JSOHlwamVQVWdJQT09)
Rui Han, LSU

A Polynomial Roth Theorem for Corners in the Finite Field Setting

The investigation of polynomial extensions of the Roth's theorem was started by Bourgain and Chang, and has seen a lot of recent advancements. The most striking of these are a series of results of Peluse and Prendiville which prove quantitative versions of the polynomial Roth and Szemerédi theorems in the integer setting. There is yet no corresponding result for corners, the two dimensional setting for polynomial Roth's Theorem, where one considers progressions of the form (x, y), (x+t, y), (x, y+t^2) in [1 ,..., N]^2, for example.

We will talk about a recent result on the corners version of the result of Bourgain and Chang, showing an effective bound for a three term polynomial Roth's theorem in the finite field setting. This is based on joint work with Michael Lacey and Fan Yang.

Posted February 14, 2021

Last modified April 18, 2021

Applied Analysis Seminar Questions or comments?

3:30 am - 4:30 am Zoom (Link: https://lsu.zoom.us/j/99328865552?pwd=TThEaXF0cjQzVFprYk1ENGc2UmxGdz09)
Marcelo Disconzi, Department of Mathematics, Vanderbilt University

General-relativistic viscous fluids

The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical relativity simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and comprehensive theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem, discuss the mathematics behind it, and report on a new approach to relativistic viscous fluids that addresses these issues.

Posted August 23, 2021

Last modified October 6, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom link: https://lsu.zoom.us/j/5494314978
Jun-cheng Wei, University of British Columbia
Canada Research Chair (CRC Tier I) in Nonlinear Partial Differential Equations

Stability of Sobolev Inequalities and related topics

Suppose $u\in \dot{H}^1(\mathbb{R}^n)$. In 1984, Struwe proved that if $||\Delta u+u^{\frac{2n}{n-2}}||_{H^{-1}}:=\Gamma(u)\to 0$ then $\delta(u)\to 0$, where $\delta(u)$ denotes the $\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. In 2020, Figalli and Glaudo obtained the first quantitative version of Struwe's decomposition in lower dimensions, namely $\delta(u)\lesssim \Gamma(u)$ when $3\leq n\leq 5$. In this talk, I will present an optimal nonlinear estimate: $\delta (u)\leq C\Gamma(u)|\log \Gamma(u)|^{\frac{1}{2}}$ if $n=6$ and $\delta (u)\leq C |\Gamma(u)|^{\frac{n+2}{2(n-2)}}$ if $n\geq 7.$ Related stability questions for isoperimetric inequality and harmonic map inequality will be discussed. (Joint work with B. Deng and L. Sun.)

Posted October 5, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom
Mihaela Ignatova, Temple University

Electroconvection in Fluids

We describe results on an electroconvection model in fluids. The model consists of two dimensional Navier-Stokes equations, driven by electrical and body forces, coupled to an advection and fractional diffusion equation for the surface charge density, driven by voltage applied at the boundary. We prove global regularity of solutions and show that the long-time behavior is described by a finite dimensional attractor. In the absence of body forces, the attractor reduces to a singleton, i.e., there is a unique, globally stable stationary solution.

Posted October 18, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
Frederic Marazzato, Louisiana State University

Variational Discrete Element Methods

Discrete Element Methods (DEM) have been introduced in [Hoover et al, 1974] to compute granular materials. Their application to compute elastic materials has remained an open question for a long time [Jebahi et al, 2015]. A first step in that direction was achieved in [Monasse et al, 2012], however the method suffered from several limitations. In [Marazzato et al, 2020], a discretization method for dynamic elasto-plasticity was proposed based on DEM by making a link with hybrid finite volume methods. Only cell dofs are used and a reconstruction is devised to obtain P^1 non-conforming polynomials in each cell and thus constant strains and stresses in each cell. An adaptation of the method consisting in adding cellwise constant rotational dofs made possible the computation of Cosserat materials [Marazzato, 2021]. Taking advantage of the capacity of DEM to deal with discontinuous displacement fields, another adaptation of the method made possible the computation of fracture in two-dimensional settings. Numerical examples for both static and dynamic computations in two and three dimensions will demonstrate the robustness of the proposed methodology.

Posted October 8, 2021

Last modified October 18, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/5494314978
Stefan Steinerberger, University of Washington

Laplacian Eigenfunctions: Hot Spots and Anti Hot Spots

The Hot Spots conjecture (first posed by Rauch in the 1970s) is a particularly fun open problem: vaguely put, if you let the heat equation act for a long period of time in an insulated room then, for generic initial data, the hottest and the coldest spot are both on the boundary of the room. I will discuss the origin behind the problem and survey some of the existing results. However, as first shown by Burdzy and Werner around 20 years ago, the Hot Spots conjecture fails in certain selected domains (very curious domains, I will show many pictures). However, it cannot fail too much: for all domains in all dimensions, there is a universal inverse result and the hottest spot inside the domain is at most 60 times as hot as the hottest spot on the boundary.

Posted September 5, 2021

Last modified November 3, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/93759214365
Christopher Henderson, University of Arizona

Pushed, pulled, and pushmi-pullyu fronts for the Burgers-FKPP equation: stability and long-time asymptotics

A minimal model for flame propagation in a fluid is the Burgers-FKPP equation. This reaction-advection-diffusion model involves a parameter beta measuring the strength of the induced drift, and a major question is how the fluid dynamics affect the long-time behavior of solutions. By studying special `traveling wave' solutions of this equation, consisting of a fixed profile that moves at a constant positive speed, it has long been known that there are two regimes: (1) when beta is less than 2, fronts are `pulled' by their behavior at infinity, and (2) when beta is greater than 2, fronts are `pushed' by the behavior at the front. In essence, regime (1) involves studying a linear problem (albeit on a noncompact set), while regime (2) involves truly nonlinear analysis, although essentially only on a compact set. However, the phase-plane analysis used to establish this is unable to say anything about the long-time behavior of generic solutions to the Burgers-FKPP equation. In particular, the stability of these traveling waves was unknown. This talk will discuss a recent work with An and Ryzhik in which we establish the precise long-time dynamics of the traveling waves, including showing their stability. Surprisingly, the proof is extremely intricate. A particularly complex case, which will be the main focus of the talk, is beta = 2, when the noncompactness of the pulled case is present with the nonlinearity of the pushed case. The analysis of this case involves techniques not usually seen applied to such problems, such as relative entropy arguments.

Posted September 22, 2021

Last modified November 12, 2021

Applied Analysis Seminar Questions or comments?

9:30 am - 10:30 am Zoom: https://lsu.zoom.us/j/93122784507
Thomas Alazard, École Normale Supérieure Paris-Saclay, CNRS

Entropies of free surface flows in fluid dynamics

I will discuss recent works with Didier Bresch, Nicolas Meunier, and Didier Smets on the dynamics of a free surface carried by an incompressible flow obeying Darcy's law. This talk focuses on monotonicity properties of different kinds: maximum principles, Lyapunov functions, and entropies. The analysis is based on exact identities which in turn allow us to study the Cauchy problem.

Posted October 1, 2021

Last modified November 21, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 232
Fan Yang, LSU

Improving estimates for discrete averages

In this talk I will discuss some recent results on discrete l^p improving estimates for averages along the prime numbers and polynomials. We will show how the Hardy-Littlewood Circle method can be used to prove the first sharp results for square integers and prime numbers, and how the general polynomial average is connected to the Vinogradov's mean value theorem. This talk is based on joint works with Rui Han, Vjekoslav Kovac (University of Zagreb), Ben Krause (KCL), Michael Lacey (Gatech) and Jose Madrid (UCLA).

Posted October 26, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom Webinar
Debdeep Bhattacharya, Mathematics Department, Louisiana State University

Long-time behavior of low regularity data in the 2d modified Zakharov-Kuznetsov equation

Posted November 10, 2021

Last modified December 5, 2021

Applied Analysis Seminar Questions or comments?

3:30 pm https://lsu.zoom.us/j/8706058864
Burak Hatinoglu, UC Santa Cruz

Spectral Properties of Periodic Elastic Beam Lattices

This talk will be on the spectral properties of elastic beam Hamiltonian defined on periodic hexagonal lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar valued fourth-order Schrödinger operator equipped with a real periodic symmetric potential. Unlike the second-order Schrödinger operator commonly applied in quantum graph literature, here the self-adjoint vertex conditions encode geometry of the graph by their dependence on angles at which edges are met. I will firstly consider this Hamiltonian on a special equal-angle lattice, known as graphene or honeycomb lattice. I will also discuss spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a recent joint work with Mahmood Ettehad (University of Minnesota), https://arxiv.org/pdf/2110.05466.pdf.

Posted January 6, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
Marta Lewicka, University of Pittsburgh

Geodesics and isometric immersions in kirigami

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. We are concerned with two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the full family of polygonal geodesics can be simultaneously rectified into a straight line via a piecewise affine planar isometric immersion.

Posted January 14, 2022

Last modified February 8, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
Petronela Radu, University of Nebraska, Mathematics Department

Nonlocal operators to the boundary and beyond

The emergence of nonlocal theories as promising models in different areas of science (continuum mechanics, biology, image processing) has led the mathematical community to conduct varied investigations of systems of integro-differential equations. In this talk I will present some results we obtained at the operator level (including a new formulation of nonlocal Calculus with corresponding Helmholtz-Hodge type decompositions) as well as for systems of integral equations with weakly singular kernels (conservation laws, diffusion equations), flux-type boundary conditions with applications at both, theoretical, and applied levels.

Posted January 16, 2022

Last modified March 4, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 am Zoom: https://lsu.zoom.us/j/5494314978
Chenjie Fan, Academy of Mathematics and Systems Science of the Chinese Academy of Sciences

On stochastic NLS: wellposedness and long time behavior

We present our study on stochastic NLS. The aim is to understand how a noise can impact a dispersive system. We will start with local theory, talk about global wellposedness, and report our recent work on long time behavior. Joint work with Weijun Xu and Zehua Zhao.

Posted January 17, 2022

Last modified February 10, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm
Jonas Lührmann, Texas A&M University

Asymptotic stability of the sine-Gordon kink under odd perturbations

The sine-Gordon model is a classical nonlinear scalar field theory that was discovered in the 1860s in the context of the study of surfaces with constant negative curvature. Its equation of motion features soliton solutions called kinks and breathers, which play an important role for the long-time dynamics. I will begin the talk with an introduction to classical 1D scalar field theories and the asymptotic stability problem for kinks. After surveying recent progress on the problem, I will present a joint work with W. Schlag on the asymptotic stability of the sine-Gordon kink under odd perturbations. Our proof is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key aspects are a super-symmetric factorization property of the linearized operator and a remarkable non-resonance property of a variable coefficient quadratic nonlinearity.

Posted January 17, 2022

Last modified March 27, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom
Stan Palasek, UCLA

Quantitative regularity theory for the Navier-Stokes equations in critical spaces

An important question in the theory of the incompressible Navier-Stokes equations is whether boundedness of the velocity in various norms implies regularity of the solution. Critical norms are conjectured to be (roughly) the threshold between positive and negative answers to this question. Of particular interest are 3D solutions in the critical endpoint space $L_t^\infty L_x^3$ for which Escauriaza-Seregin-Sverak famously proved global regularity. Recently Tao improved upon this result by proving quantitative bounds on the solution and conditions on a hypothetical blowup. In this talk we discuss the quantitative approach to regularity including some sharper results in the axisymmetric case, as well as extensions to other critical spaces and to higher dimensions.

Posted January 18, 2022

Last modified March 31, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/5494314978
Mariana Smit Vega Garcia, Western Washington University

Almost minimizers for obstacle problems

In the applied sciences one is often confronted with free boundaries, which arise when the solution to a problem consists of a pair: a function u (often satisfying a partial differential equation), and a set where this function has a specific behavior. Two central issues in the study of free boundary problems are: (1) What is the optimal regularity of the solution u? (2) How smooth is the free boundary? The study of the classical obstacle problem - one of the most renowned free boundary problems - began in the ’60s with the pioneering works of G. Stampacchia, H. Lewy, and J. L. Lions. During the past decades, it has led to beautiful developments, and its study still presents very interesting and challenging questions. In contrast to the classical obstacle problem, which arises from a minimization problem (as many other PDEs do), minimizing problems with noise leads to the notion of almost minimizers. In this talk, I will introduce obstacle-type problems and overview recent developments in almost minimizers for the thin obstacle problem, illustrating techniques that can be used to tackle questions (1) and (2) in various settings.

Posted March 28, 2022

Last modified March 30, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 232
Frank Sottile, Texas A&M

Critical Points of Discrete Periodic Operators via Toric Varieties

It is believed that the dispersion relation of a Schrodinger operator with a periodic potential has non-degenerate critical points. In work with Kuchment and Do, we considered this for discrete operators on a periodic graph G, for then the dispersion relation is an algebraic hypersurface. A consequence is a dichotomy; either almost all parameters have all critical points non-degenerate or almost all parameters give degenerate critical points, and we showed how tools from computational algebraic geometry may be used to study the dispersion relation. \[ \hspace{1em} \] With Matthew Faust, we use ideas from combinatorial algebraic geometry to give an upper bound for the number of critical points at generic parameters, and also a criterion for when that bound is obtained. The dispersion relation has a natural compactification in a toric variety, and the criterion concerns the smoothness of the dispersion relation at toric infinity.

Posted March 30, 2022

Applied Analysis Seminar Questions or comments?

3:30 am - 4:30 am Lockett Hall 232
Davit Harutyunyan, EPFL

On the extreme rays of the cone of 3 by 3 quasiconvex quadratic forms

The extreme rays of the convex cone of 3 by 3 quasiconvex quadratic forms play an important role in applied mathematics and in particular in the theory of composite materials. In this work, we provide a characterization of 3 by 3 quasiconvex quadratic forms, the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. While the problem arises in Applied Mathematics (The Theory of Composites), it is also related to the problem of "Sums of Squares" in Convex Geometry and Real Algebraic Geometry. We combine methods from classical Linear Algebra, Convex Geometry and Real Algebraic Geometry in the proofs. This is joint work with Narek Hovsepyan.

Posted January 20, 2022

Last modified February 8, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
Davit Harutyunyan, University of California Santa Barbara

TBA

Posted February 18, 2022

Last modified April 17, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/5494314978
Pablo Raul Stinga, Iowa State University

Regularity for C^{1,a} interface transmission problems

We present existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with C^{1,a} interfaces. The main tool we develop for the regularity estimates is a new geometric stability argument based on the mean value property. This is joint work with Luis A. Caffarelli (UT Austin) and our graduate student María Soria-Carro (UT Austin).

Posted February 4, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233TBA

Posted February 4, 2022

Last modified November 29, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett 233
Armin Schikorra, University of Pittsburgh

On Calderón–Zygmund type estimates for nonlocal PDE

I will report on progress obtained for the $W^{s,p}$-regularity theory for nonlocal/fractional equations of differential order $2s$ with bounded measurable Kernel. Namely, under (not yet optimal) assumptions on the kernel we obtain $W^{t,p}$-estimates for suitable right-hand sides, where $2s>t>s$. Technically we compare such equations via a commutator estimate to a simpler fractional equation. Based on joint works with M.M Fall, T. Mengesha, S. Yeepo.

Posted August 15, 2022

Last modified September 16, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Zoom Meeting
Shibin Dai, University of Alabama

Degenerate Diffusion and Interface Motion of Single Layer and Bilayer Structures

Degenerate diffusion plays an important role in the interface motion of complex structures. The degenerate Cahn-Hilliard equation is a widely used model for single layer structures. It has been commonly believed that degenerate diffusion eliminates diffusion in the bulk phases and results in surface diffusion only. We will show that due to the curvature effect there is porous medium diffusion in the bulk phases, and the geometric evolution of single layer structures is mediated by the porous medium diffusion process. We will also discuss the existence of weak solutions for the degenerate CH equation. For bilayer structures the Functionalized Cahn-Hilliard (FCH) equation is a new model that has been extensively studied in recent years. We will discuss the existence and nonnegativity of weak solutions for the degenerate FCH equation, and the corresponding interface motions.

Posted April 25, 2022

Last modified September 22, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom: https://lsu.zoom.us/j/91528952709?pwd=UmRUQ1Rxb2IvS2l6M0l4MlMxbG15Zz09
Hao Jia, School of Mathematics, University of Minnesota

Uniform linear inviscid damping near monotonic shear flows in the whole space

In recent years tremendous progress was made in understanding the ``inviscid damping" phenomenon near shear flows and vortices, which are steady states for the 2d incompressible Euler equation, especially at the linearized level. However, in real fluids viscosity plays an important role. It is natural to ask if incorporating the small but crucial viscosity term (and thus considering the Navier Stokes equation in a high Reynolds number regime instead of Euler equations) could change the dynamics in any dramatic way. It turns out that for the perturbative regime near a spectrally stable monotonic shear flows in an infinite periodic channel (to avoid boundary layers and long wave instabilities), we can prove uniform-in-viscosity inviscid damping. The proof introduces techniques that provide a unified treatment of the classical Orr-Sommerfeld equation in a way analogous to Rayleigh equations.

Posted January 25, 2022

Last modified October 4, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett Hall 233
Kasso Okoudjou, Tufts University

Topics in analysis on fractals and self-similar graphs

The goal of this talk is to present some topics related to what is known as analysis on fractal sets such as the Sierpinski gasket. This theory is based on the spectral analysis of a corresponding Laplace operator which we will introduce in the first part of the talk. We will then review certain fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schr\"odinger operators, and the theory of orthogonal polynomials. In the last part of the talk, I will introduce a self-similar analog of the almost Mathieu operators (AMO), and present some results pertaining to their spectral properties. These results are obtained by using the so-called spectral decimation method which is one of the important tools in the spectral analysis of fractal Laplacians.

Posted October 4, 2022

Last modified October 7, 2022

Applied Analysis Seminar Questions or comments?

4:10 pm Lockett Hall 233
Matthew Faust, TX A&M University

Reducibility of Bloch and Fermi varieties via discrete geometry

Given an infinite ZZ^n periodic graph G, the Schrödinger operator acting on G is a graph Laplacian perturbed by a potential at every vertex. Complexifying and choosing a potential periodic to a full rank subgroup of ZZ^n fixes a representation of the operator as a finite matrix whose entries are Laurent polynomials. The vanishing set of the characteristic polynomial of this matrix yields the Bloch variety, and the vanishing set for a fixed eigenvalue gives the Fermi variety. We will focus our attention on the reducibility of these varieties. Understanding the reducibility of Bloch and Fermi varieties is important in the study of the spectrum of periodic operators, providing insight into the structure of spectral edges, embedded eigenvalues, and other applications. In this talk we will present several new criteria for determining when Bloch and Fermi varieties are irreducible for infinite families of discrete periodic operators. This is joint work with Jordy Lopez.

Posted September 21, 2022

Last modified October 17, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 233
Doosung Choi, LSU

Inverse problem in potential theory using Faber polynomials

This presentation concerns the inverse problem of determining a planar conductivity inclusion. We analytically reconstruct from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements. The primary result is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion.

Posted October 3, 2022

Last modified October 31, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/5494314978
Donatella Danielli, Arizona State University

Regularity properties in obstacle-type problems for higher-order fractional powers of the Laplacian

In this talk we will discuss a sampler of obstacle-type problems associated with the fractional Laplacian. Our goals are to establish regularity properties of the solution and to describe the structure of the free boundary. To this end, we combine classical techniques from potential theory and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas. This is joint work with A. Haj Ali (Arizona State University) and A. Petrosyan (Purdue University).

Posted September 22, 2022

Last modified November 11, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom: https://lsu.zoom.us/j/92890365678?pwd=SlJmN0JnQnRvTFYxd1QyUjhqR0tqQT09
Arghir Zarnescu, Basque Center for Applied Mathematics

On the motion of several small rigid bodies in a viscous incompressible fluid

We consider the motion of N rigid bodies contained in a domain in dimension two or three. We show the fluid flow is not influenced by the presence of the bodies in the asymptotic limit as the size of the bodies tends to zero. The result depends solely on the geometry of the bodies and is independent of their mass densities. Collisions are allowed and the initial data are arbitrary with finite energy. This is joint work with Eduard Feireisl and Arnab Roy.

Posted July 7, 2022

Last modified November 20, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom: https://lsu.zoom.us/j/8811458211?pwd=Um1DV3J6YkFSbkkzSldwSXU1cFJqQT09
Thomas Chen, University of Texas at Austin

On the emergence of a quantum Boltzmann equation near a Bose-Einstein condensate

The mathematically rigorous derivation of nonlinear Boltzmann equations from first principles in interacting physical systems is an extremely active research area in Analysis, Mathematical Physics, and Applied Mathematics. In classical physical systems, rigorous results of this type have been obtained for some models. In the quantum case on the other hand, the problem has essentially remained open. In this talk, I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics around a Bose-Einstein condensate, within the quantum field theoretic description of an interacting Boson gas. This is based on joint work with Michael Hott.

Posted October 2, 2022

Last modified November 29, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
James Scott, Columbia University

Geometric Rigidity Theorems for Nonlocal Continuum Theories of Linear and Nonlinear Elasticity

We present several quantitative results that generalize known nonlocal rigidity relations for vector fields representing deformations of elastic media. We show that the distance in Lebesgue norm of a deformation from a rigid motion is bounded by a multiple of a strain energy associated to the deformation. This nonconvex energy is a nonlocal constitutive relation that represents the extent to which the deformation stretches and shrinks distances. This inequality can be thought of as a nonlinear fractional Poincaré-Korn inequality. We linearize this inequality to obtain a fractional Poincaré-Korn inequality for Lipschitz domains with an explicit universal bounding constant. This inequality is also valid for more general interaction kernels of non-fractional type, which we demonstrate by using a compactness argument. We apply these inequalities to obtain quantitative statements for solutions to variational problems arising in peridynamics, dislocation models, and phase transition dynamics.

Posted November 10, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom
Jeffrey Rauch, University of Michigan

Earnshaw’s Theorem in Electrostatics

This result dating to 1842 asserts that a charge in a static electrostatic field can never be in a stable equilibrium. In spite of many partial results a complete proof was first given in 1987. The present talk concerns generalizations from Section 116 of Maxwell’s treatise. There Maxwell explains (but does not prove) why a rigid charged body or a perfect conducting body or a dielectric body in a static field can never be in a stable equilibrium. We prove the result for conductors and dielectrics. The charged rigid body remains open. This joint work with G. Allaire appeared in the Archive for Rational Mechanics in 2017.

Posted October 12, 2022

Last modified November 9, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Lockett Hall 233 and Zoom
Yue Yu, Lehigh University

Learning Nonlocal Neural Operators for Complex Physical System Modeling

For many decades, physics-based PDEs have been commonly employed for modeling complex system responses, then traditional numerical methods were employed to solve the PDEs and provide predictions. However, when governing laws are unknown or when high degrees of heterogeneity present, these classical models may become inaccurate. In this talk we propose to use data-driven modeling which directly utilizes high-fidelity simulation and experimental measurements to learn the hidden physics and provide further predictions. In particular, we develop PDE-inspired neural operator architectures, to learn the mapping between loading conditions and the corresponding system response. By parameterizing the increment between layers as an integral operator, our neural operator can be seen as the analog of a time-dependent nonlocal equation, which captures the long-range dependencies in the feature space and is guaranteed to be resolution-independent. Moreover, when applying to (hidden) PDE solving tasks, our neural operator provides a universal approximator to a fixed point iterative procedure, and partial physical knowledge can be incorporated to further improve the model’s generalizability and transferability. As an application, we learn the material models directly from digital image correlation (DIC) displacement tracking measurements on a porcine tricuspid valve leaflet tissue, and show that the learnt model substantially outperforms conventional constitutive models.

Posted January 8, 2023

Last modified February 12, 2023

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3:30 pm Zoom
Justin Holmer, Brown University

Well/Ill-posedness of the Boltzmann equation with constant collision kernel

Drawing upon methods from the field of nonlinear dispersive PDEs, we investigate well/ill-posedness for the 3D Boltzmann equation with constant collision kernel in the Sobolev spaces. We find that the threshold space is one-half derivative above the scale invariant space, and prove ill-posedness below this threshold by constructing a family of special solutions, which are neither near equilibrium nor self-similar, and exhibit a "norm deflation" behavior -- a rapid drop in the Sobolev norm that breaks the uniform continuity of the data-to-solution map. This is joint work with Xuwen Chen (University of Rochester)

Posted February 10, 2023

Last modified March 3, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/5494314978?pwd=SmpvVDRpaFY2dGxqcGlIT0kxTzVMdz09
Zihui Zhao, University of Chicago

Counter-example in boundary unique continuations

Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular set. In this talk, I will talk about some boundary unique continuation results, and show that these results are sharp by giving explicit examples using harmonic measures. This is joint work with C. Kenig

Posted February 20, 2023

Last modified March 3, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/5494314978?pwd=SmpvVDRpaFY2dGxqcGlIT0kxTzVMdz09
Max Engelstein, University of Minnesota

On global graphical solutions to free boundary problems

The Bernstein problem for minimal surfaces asks whether a globally defined minimal hypersurface given by the graph of a function in dimension $n$ must be a hyperplane. This was resolved by the combined work of De Giorgi, Simons and then De Giorgi-Bombieri-Giusti; showing that the answer is yes when $n \leq 8$ and no when $n\geq 9$. In this talk we will discuss recent progress towards this question for one-phase free boundary problems of Bernoulli type. This is joint with Xavier Fernandez-Real (EPFL) and Hui Yu (NUS).

Posted February 1, 2023

Last modified March 20, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 232
Kirill Cherednichenko, University of Bath

Operator-norm homogenisation for Maxwell equations on periodic singular structures

I will discuss a new approach to obtaining uniform operator asymptotic estimates in periodic homogenisation. Based on a novel uniform Poincaré-type inequality, it bears similarities to the techniques I developed with Cooper (ARMA, 2016) and Velcic (JLMS, 2022). In the context of the Maxwell system, the analytic framework I will present leads to a new representation for the asymptotics obtained by Birman and Suslina in 2007 for the full system and by Suslina in 2004 for the electric field in the presence of currents. As part of the new asymptotic construction, I will link the leading-order approximation to a family of "homogenised" problems, which was not possible using the earlier method. The analysis presented applies to a class of inhomogeneous structures modelled by arbitrary periodic Borel measures. However, the results are new even for the particular case of the Lebesgue measure. This is joint work with Serena D'Onofrio.

Posted February 28, 2023

Last modified April 23, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom: https://lsu.zoom.us/j/5494314978
Hung Tran, University of Wisconsin Madison

Periodic homogenization of Hamilton-Jacobi equations: some recent progress.

I first give a quick introduction to front propagations, Hamilton-Jacobi equations, and homogenization theory. I will then show that the optimal rate of convergence is $O(\varepsilon)$ in the convex setting and some nonconvex cases. I will also mention finer results on the effective fronts in two dimensions. Connections to stable norms in Riemannian geometry will also be made. Based on various joints work with W. Jing and Y. Yu.

Posted August 20, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm Lockett 232
Fabrice Baudoin, University of Connecticut

Dirichlet forms on metric measure spaces as Mosco limits of Korevaar-Schoen energies

We will give sufficient general conditions for the existence of Mosco limits of Korevaar-Schoen L2 energies, first in the context of Cheeger spaces and then in the context of fractal-like spaces with walk dimension greater than 2. Among the ingredients, a new Rellich- Kondrachov type theorem for Korevaar-Schoen-Sobolev spaces is of independent interest. The talk will be based on a joint work with Patricia Alonso-Ruiz (TAMU)

Posted August 1, 2023

Last modified September 10, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom (https://lsu.zoom.us/j/96065057555)
Gautam Iyer, Carnegie Mellon University

Using mixing to accelerate convergence of Langevin systems, and applications to Monte Carlo methods

A common method used to sample from a distribution with density proportional to $p = e^{-V/\kappa}$ is to run Monte Carlo simulations on an overdamped Langevin equation whose stationary distribution is also proportional to $p$. When the potential $V$ is not convex and the temperature $\kappa$ is small, this can take an exponentially large (i.e. of order $e^{C/\kappa}$) amount of time to generate good results. I will talk about a method that introduces a "mixing drift" into this system, which allows us to rigorously prove convergence in polynomial time (i.e. a polynomial in $1/\kappa$). This is joint work with Alex Christie, Yuanyuan Feng and Alexei Novikov.

Posted September 25, 2023

Applied Analysis Seminar Questions or comments?

9:00 am Zoom (email dmassatt@lsu.edu for link)TBA

Posted September 5, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom
Camil Muscalu, Cornell University

TBA

Posted December 9, 2022

Last modified September 5, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 233 and Zoom
Nicolas Meunier, LaMME, Universite Evry Val D'Essonne

TBA

Posted September 1, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm Zoom
Amir Sagiv, Technion Israel Institute of Technology

TBA

Posted September 20, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 233
Burak Hatinoglu, Michigan State University

TBA on spectral theory of quantum graphs

(Host: Stephen Shipman)

Posted September 6, 2023

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3:30 pm - 4:30 pm Zoom
Ivan Veselić, Technical University of Dortmund

TBA

Posted September 3, 2023

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3:30 pm - 4:30 pm Zoom
Yannick Sire, Johns Hopkins University

TBA

Posted September 3, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm ZoomTBA

Posted September 22, 2023

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232
Eduard-Wilhelm Kirr, University of Illinois Urbana-Champagne

TBA

Posted September 4, 2023

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3:30 pm - 4:30 pm Zoom
Piermarco Cannarsa, Università degli studi di Roma "Tor Vergata"

Generalised characteristics of Hamilton-Jacobi equations, propagation of singularities, and long-time behaviour

A generalised characteristic (GC) is a solution of certain differential inclusions that play a crucial role for propagation of singularities of solutions to Hamilton-Jacobi equations $H(x,u(x),Du(x))=0$. GC's were introduced in (D1977), in the context of hyperbolic conservation laws and then adapted to Hamilton-Jacobi equations in (AC2002) and (CY2009). In this talk, we will discuss several topics related to GC's including restricted classes of characteristics introduced in (KS2016), uniqueness issues, continuation properties in connection with propagation of singularities (CC2017). Then, for mechanical systems on the torus, that is, \begin{equation*} \frac 12|Du(x)|^2+V(x)=\alpha \\end{equation*} we will study the long time behaviour of GC's. By using limiting occupational measures, we will show that the critical set of $u$ is an approximate attractor for the GC flow. We will also give a criterion to decide whether, asymptotically, a GC is almost surely either singular or arbitrarily close to the regular critical set of $u$. \bigskip {\bf\large References} \medskip \begin{itemize} \item[(AC2002)] Albano, P., Cannarsa, P.: Propagation of singularities for solutions of nonlinear first order partial differential equations. Arch. Ration. Mech. Anal. 162(1), 1-23 (2002) \item[(CC2017)] Cannarsa, P., Cheng, W.: Generalized characteristics and Lax-Oleinik operators: global theory. Calc. Var. Partial Differ. Equ. 56(5), 31 (2017) \item[(CY2009)] Cannarsa, P., Yifeng, Yu.: Singular dynamics for semiconcave functions. J. Eur. Math. Soc. (JEMS) 11(5), 999-1024 (2009) \item[(D1977)] Dafermos, C.M.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J. 26(6), 1097-1119 (1977) \item[(KS2016)] Khanin, K., Sobolevski, A.: On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 219(2), 861-885 (2016) \end{itemize}