Stability of the equilibria

for spatially periodic flows in porous media

Joachim Escher Georg Prokert

Universität Gesamthochschule Kassel Fachbereich 17 Mathematik/Informatik Heinrich-Plett-Str. 40, 34109 Kassel

                     escher@mathematik.uni-kassel.de
                     prokert@mathematik.uni-kassel.de

Abstract

We prove exponential stability of the equilibria for two moving boundary problems describing flows in porous media which are of Hele-Shaw and Stefan type, respectively. The main tool is the principle of linearized stability for fully nonlinear parabolic evolution equations. Crucial points are the assumption of spatial periodicity for the flows and the identification of conserved quantities.

Key words: fully nonlinear parabolic equation, exponential stability, moving interface

1 Introduction

The investigation of various mathematical models for flows in porous media has recently attracted a lot of interest from different points of view. This article is concerned with two such models that share the following basic properties:

• the flow is governed by Darcy’s law,

• the porous medium is either completely saturated by the liquid or dry, and both phases are separated by a sharp interface,

• the main interest is in the determination of the motion of this interface, i.e. we are concerned with a moving boundary problem.

The use of Darcy’s law together with linear constitutive equations leads to linear governing equations for the velocity potential u. From a mathematical point of view, the moving boundary problems that are formed by these equations together with appropriate initial and boundary conditions are equivalent to a (multidimensional) Hele-Shaw flow problem and a Stefan-type problem, respectively. We will consider a geometric setting and driving forces describing an infinite layer of liquid which moves under the influence of gravity; no other driving forces are considered. Moreover, the liquid domain is bounded from below by a fixed impermeable horizontal bottom. The corresponding free boundary problems have been discussed in [7, 8, 10, 11] for the case of an incompressible porous medium (the Hele-Shaw problem) and in [6] in the case of a compressible porous medium (the Stefan-type problem, see also [4, 5]).

Our analysis is based on the methods used in [6, 7, 8] which can be briefly and informally sketched as follows:

1. The problem is reformulated as a nonlinear initial boundary value problem on a fixed reference domain, using a transformation via an a priori unknown, timedependent diffeomorphism.

2. By discussion of the linearization, this problem is identified as an abstract parabolic Cauchy problem.

3. Well-posedness of the problem and smoothness of its solution can be shown using the theory of maximal regularity for fully nonlinear parabolic equations, see e.g. [2, 3, 12].

In general, these methods can also be used to investigate the stability of stationary solutions on the basis of the “principle of linearized stability”, i.e. by studying the spectrum of the linearized evolution equation near a stationary solution.

However, even in the simplest two-dimensional case of the Hele-Shaw type problem it turns out that the L^{2}-realization of the corresponding linearized evolution equation has a pure continuous spectrum, given by [0, ∞). Moreover, there is a continuous family of equilibria given by all liquid layers of constant density.

It is the aim of this article to prove exponential stability for a modified version of the two moving boundary value problems above, overcoming these difficulties in the following way: By demanding periodicity of the flows in space, the unbounded domain ℝ^{n} is replaced by the torus 𝕋^{n}, and by the identification of a conserved quantity, the evolution is a priori restricted to a submanifold of the phase space which intersects the set of equilibria in a single point. It is shown that the equilibrium of these restrictions is exponentially stable.

2 Flows in incompressible porous media

In the description of the problem as well as in the notation, we closely follow [8]. Let α ∈ (0, 1) and define

U := {f ∈ BUC^{2+α}(ℝ^{n}) | inf_{x∈ℝ^{n}} f(x) > 0} .

Given f ∈ U, let

Ω̃_{f} := {(x, y) ∈ ℝ^{n} × ℝ | 0 < y < f(x), x ∈ ℝ^{n}}

and denote the components of its boundary by

Γ̃_{f} := {(x, f(x)) | x ∈ ℝ^{n}}, Γ̃_{0} := ℝ^{n} × {0}.

We consider the elliptic boundary value problem (BVP)

\begin{matrix}Δ_{n+1}u & = & 0 & in Ω̃_{f}, \\ ∂_{n+1}u & = & 0 & on Γ̃_{0}, \\ u & = & y & on Γ̃_{f}, \end{matrix} \}  (2.1)

where Δ_{n+1} is the (n + 1)-dimensional Laplacian, ∂_{n+1} denotes the (n + 1)st partial derivative and y is the (n + 1)st coordinate function on ℝ^{n+1} ≅ ℝ^{n} × ℝ. This BVP is a simple model for the motion of an incompressible liquid in a rigid porous medium under the influence of gravity. The liquid occupies the domain Ω̃_{f} over the impermeable bottom Γ̃_{0}, and u is the velocity potential. (The well-posedness of this problem will be shown under additional assumptions later.) One is interested in the evolution of the interface separating the saturated from the dry part of the porous medium which is represented by the boundary component Γ̃_{f(t)}, where now f : [0, ∞) ⟶ U. This evolution is described by the nonlinear Cauchy problem

\begin{matrix}∂_{t}f(t, ·) + \sqrt{1 + |∇_{n}f(t, ·)|^{2}}∂_{ν}u(·, f(t, ·)) & = & 0, \\ f(0, ·) & = & f_{0} \end{matrix} \}  (2.2)

on ℝ^{n}, where f_{0} corresponds to the initial interface, ∇_{n} denotes the gradient on ℝ^{n}, u is the solution of (2.1), and ∂_{ν}u denotes the derivative of u with respect to the outer unit normal on Γ̃_{f}.

We remark that problem (2.1), (2.2) is mathematically equivalent to a multidimensional one-phase Hele-Shaw problem with mixed boundary conditions.

In our stability analysis we shall make the following crucial periodicity requirement on f and on u:

f(t, x + 2kπe_{j}) = f(t, x) ∀x ∈ ℝ^{n}, j = 1, …, n, k ∈ ℤ, t ≥ 0  (2.3)
u(x + 2kπe_{j}, y) = u(x, y) ∀(x, y) ∈ Ω̃_{f(t)}, j = 1, …, n, k ∈ ℤ, t ≥ 0. (2.4)

In other words, instead of (2.1), (2.2) we discuss

\begin{matrix}Δ_{n+1}u & = & 0 & in Ω_{f(t)}, & t ≥ 0, \\ ∂_{n+1}u & = & 0 & on Γ_{0}, & t ≥ 0, \\ u & = & y & on Γ_{f(t)}, & t ≥ 0, \\ ∂_{t}f(t, ·) + \sqrt{1 + |∇_{n}f(t, ·)|^{2}}∂_{ν}u(·, f(t, ·)) & = & 0, & on 𝕋^{n}, & t > 0, \\ f(0, ·) & = & f_{0} & on 𝕋^{n} \end{matrix} \}  (2.5)

where

Ω_{f(t)} := {(x, y) ∈ 𝕋^{n} × ℝ | 0 < y < f(t, x), x ∈ 𝕋^{n}},
Γ_{f(t)} := {(x, f(t, x)) | x ∈ 𝕋^{n}}, Γ_{0} := 𝕋^{n} × {0},

for t ≥ 0 and 𝕋^{n} denotes the n-dimensional torus. For the sake of simplicity, we identify periodic functions on ℝ^{n} and functions on 𝕋^{n}. Remark: From a modeling point of view, it is unsatisfactory to demand periodicity both for f and for u. Actually, it seems reasonable to conjecture that (2.3) and u ∈ BUC^{2+α}(Ω̃_{f}) imply (2.4) . To show this, however, one needs to know whether the solution of (2.1) is unique in BUC^{2+α}(Ω̃_{f}). Unfortunately, we are only able to prove this under the additional assumption that f is constant or n = 1.

As in [7, 8], problem (2.5) is transformed to a problem on a fixed reference manifold. Let Ω := 𝕋^{n} × (0, 1),

c := \frac{1}{(2π)^{n}} ∫_{𝕋^{n}} f_{0}(x) dx,  (2.6)

set g := f − c, and define ϕ ∈ Diff^{2+α}(Ω, Ω_{f}) by

ϕ(x, y) := ϕ_{g}(x, y) = (x, (1 − y)(c + g(x))), (x, y) ∈ Ω.

Defining the push-forward and pull-back operators induced by ϕ_{g},

\begin{matrix}ϕ_{g}^{∗} : & C(\overline{Ω}_{f}) ⟶ C(\overline{Ω}), & u ↦ u ∘ ϕ_{g}, \\ ϕ_{∗}^{g} : & C(\overline{Ω}) ⟶ C(\overline{Ω}_{f}), & v ↦ v ∘ ϕ_{g}^{−1}, \end{matrix}

we introduce the transformed operators A(g) and B_{i}(g), i = 0, 1, acting on C^{2}(\overline{Ω}) by

A(g) := −ϕ_{g}^{∗} ∘ Δ_{n+1} ∘ ϕ_{∗}^{g},
B_{i}(g)v := ϕ_{g}^{∗}(γ_{i}∇_{n+1}(ϕ_{∗}^{g}v) · n_{i}), i = 0, 1,

where γ_{0} and γ_{1} stand for the trace operators on Γ_{c+g} and Γ_{0}, respectively. Moreover, n_{0} := (−∇_{n}g, 1) and n_{1} := (0, …, 0, −1). Observe that n_{0} is normal to Γ_{c+g}. For an explicit coordinate representation we refer to [8], Lemma 2.1. Although an unbounded geometry is considered there, the expressions are formally the same an in [8].

Transformation of (2.5) to Ω and Γ_{0}, respectively, yields

\begin{matrix}A(g)v & = & 0 & in Ω × [0, ∞), \\ v & = & g + c & on Γ_{0} × [0, ∞), \\ B_{1}(g)v & = & 0 & on Γ_{1} × [0, ∞), \\ ∂_{t}g + B_{0}(g)v & = & 0 & on Γ_{0} × (0, ∞), \\ g(0) & = & g_{0}, \end{matrix} \}  (2.7)

where v := ϕ_{g}^{∗}u and g_{0} := f_{0} − c.

In the following, we will employ the so-called little Hölder spaces which are defined as follows: For k ∈ ℕ, let h^{k+α}(𝕋^{n}) denote the closure of C^{∞}(𝕋^{n}) in BUC^{k+α}(𝕋^{n}) and let h^{k+α}(Ω) denote the closure of BUC^{∞}(Ω) in BUC^{k+α}(Ω). The spaces h^{k+α}(𝕋^{n}) and h^{k+α}(Ω) are Banach algebras. Hence, by arguments analogous to the proof of Lemma 2.3 in [8], it follows that for some open neighborhood O of 0 in h^{2+α}(𝕋^{n}), we have

A ∈ C^{ω} (O, L (h^{2+α}(Ω), h^{α}(Ω))) ,
B_{i} ∈ C^{ω} (O, L (h^{2+α}(Ω), h^{1+α}(𝕋^{n}))) , i = 0, 1,

where we use the obvious identification 𝕋^{n} × {i} ≅ 𝕋^{n}, and where C^{ω} denotes analytic dependence.

Moreover, given g ∈ O, we have

(A(g), γ_{0}, B_{1}(g)) ∈ Isom (h^{2+α}(Ω), h^{α}(Ω) × h^{2+α}(𝕋^{n}) × h^{1+α}(𝕋^{n})) .  (2.8)

This can be proved analogously to Theorem 3.5 in [7], the only necessary modification being the use of the compactness of Ω to ensure that any bounded and uniformly continuous function v on Ω attains its maximum on \overline{Ω}. In view of (2.8) we can define for any g ∈ O the solution operator

T (g) := (A(g), γ_{0}, B_{1}(g))^{−1}|{0} × h^{2+α}(𝕋^{n}) × {0},

i.e., given h ∈ h^{2+α}(𝕋^{n}), we write T (g)h = v for the unique solution of the BVP

A(g)v = 0 in Ω,
v = h on Γ_{0},
B_{1}(g)v = 0 on Γ_{1}.

The same arguments as in [8], Lemma 2.5, show that

T ∈ C^{ω} (O, L (h^{2+α}(𝕋^{n}), h^{2+α}(Ω))) .

Using this operator, we can reduce (2.7) to a single nonlocal Cauchy problem on 𝕋^{n}:

\begin{matrix}∂_{t}g + Φ(g) & = & 0, \\ g(0) & = & g_{0}, \end{matrix} \}  (2.9)

where

Φ(g) := B_{0}(g)T (g)(g + c).

It follows immediately from the definitions that T (g) commutes with the operator h ↦ h + c and that B_{0}(g) vanishes on constants. Hence

Φ(g) = B_{0}(g)(T (g)g + c) = B_{0}(g)T (g)g.  (2.10)

From the analyticity of T and B_{0} we get

Φ ∈ C^{ω} (O, h^{1+α}(𝕋^{n})) .  (2.11)

The following result on the solvability of (2.9) holds:

Theorem 2.1 (Existence, uniqueness, and smoothness of solutions [7, 8])

Given g_{0} ∈ O, the Cauchy problem (2.9) has a unique maximal solution

g ∈ C(J, O) ∩ C^{ω}(J̇ × 𝕋^{n}),

where J = [0, t^{+}(g_{0})) is the maximal interval of existence.

Remark: As we are concerned here with stability of the equilibrium only, we refrain from choosing O maximally large. We refer to [7, 8] for this.

The proof of Theorem 2.1 can be given in analogy to [7, 8], based on the theory of maximal regularity for fully nonlinear parabolic equations, see also [2, 3]. The only necessary changes are due to the fact that instead of an unbounded domain in ℝ^{n+1} we consider here a bounded (n + 1)-dimensional manifold. However, both the parameter-dependent estimates for the occurring elliptic boundary value problems and the interpolation results for little Hölder spaces can be carried over to this case without essential difficulties.

We start our discussion of the equilibrium g = 0 of (2.9) with an observation expressing essentially the conservation of volume for the domain saturated by the liquid. On L^{2}(𝕋^{n}) we introduce a continuous projection P by

P u = u − \frac{1}{(2π)^{n}} ∫_{𝕋^{n}} u dx.

Obviously, the restriction of P to h^{k+α}(𝕋^{n}) is continuous on these spaces, too. On all these spaces, ker P is given by the constants and Im P is given by the functions with mean 0. For the latter we will write L̃^{2}(𝕋^{n}), h̃^{k+α}(𝕋^{n}) etc.

Lemma 2.2 (Conservation of volume)

Given g ∈ O, we have Φ(g) ∈ h̃^{1+α}(𝕋^{n}).

Proof: Let f := g + c and denote by u the solution of (2.5)_{1}–(2.5)_{3}. By definition, we have T (g)g = ϕ_{g}^{∗}u and Gauss’ theorem and (2.10) give

∫_{𝕋^{n}} Φ(g) dx = ∫_{𝕋^{n}×{0}} B_{0}(g)T (g)g dx = ∫_{𝕋^{n}×{0}} ϕ_{g}^{∗}(γ_{0}(∇_{n+1}u) · n_{0}) dx
= ∫_{𝕋^{n}×\{0\}} ϕ_{g}^{∗} (\sqrt{1 + |∇_{n}g|^{2}} \frac{∂u}{∂n}) dx = ∫_{Γ_{f}} \frac{∂u}{∂n} dσ
= ∫_{Ω_{f}} Δu dx − ∫_{Γ_{0}} \frac{∂u}{∂n} dσ = − ∫_{Γ_{0}} ∂_{n+1}u dσ = 0.

Observe that due to (2.6) we have

g_{0} ∈ h̃^{2+α}(𝕋^{n}),  (2.12)

and due to Lemma 2.2 the evolution proceeds completely in h̃^{2+α}(𝕋^{n}).

Next, we calculate the Fréchet derivative

A := Φ′(0) ∈ L (h^{2+α}(𝕋^{n}), (h^{1+α}(𝕋^{n}))

of Φ at 0. One finds

Φ′(0)[h] = B′_{0}(0)[h]T (0)0 + B_{0}(0)T′(0)[h]0 + B_{0}(0)T (0)h
= B_{0}(0)T (0)h,

where we have used the linearity of the mappings T (0), B′_{0}(0)[h], and T′(0)[h]. More explicitly,

Φ′(0)[h] = ϕ_{0}^{∗} ( \frac{∂u}{∂n}|_{Γ_{c}}) ,

where u is the solution of

\begin{matrix}Δu & = & 0 & in 𝕋^{n} × (0, c), \\ u & = & h & on 𝕋^{n} × \{c\}, \\ \frac{∂u}{∂n} & = & 0 & on 𝕋^{n} × \{0\}, \end{matrix} \}  (2.13)

i.e. A is essentially the Dirichlet-Neumann operator on 𝕋^{n} × {c} for the Laplacian with homogeneous Neumann boundary condition on 𝕋^{n} × {0}. Due to the product structure of 𝕋^{n} × (0, c), (2.13) is solvable by separation of variables. Thereby one obtains complete information on the closure A_{L^{2}} of A to L^{2}(𝕋^{n}).

Lemma 2.3 (The linearized operator on L^{2})

The operator A_{L^{2}} has a complete orthogonal system of eigenfunctions e_{k}, k ∈ ℤ^{n}, given by

e_{k}(θ) = e^{ik·θ}, θ ∈ 𝕋^{n}.

These eigenfunctions correspond to the eigenvalues λ_{k} := |k| tanh c|k|, where |k| = |k_{1}| + … + |k_{n}|.

The proof rests on the well-known completeness of the system {e_{k} | k ∈ ℤ^{n}} in L^{2}(𝕋^{n}) and on the explicit solution of (2.13) for h = e_{k}, cf. (2.18) below. Observe that, in particular, the operators A_{L^{2}} and P commute, D(A_{L^{2}}) = H^{1}(𝕋^{n}), 0 ∈ σ(A_{L^{2}}), and all eigenvalues of A_{L^{2}} are nonnegative.

By standard arguments from functional analysis together with earlier results on A, the information on the spectrum of A_{L^{2}} can be carried over to A.

Lemma 2.4 (The linearized operator in h^{1+α})

Consider A as an unbounded, densely defined operator in h^{1+α}(𝕋^{n}) with D(A) = h^{2+α}(𝕋^{n}). The spectra of A and A_{L^{2}} coincide.

Proof: It can be shown as in the proof of Corollary 6.3 in [7] that

A ∈ H (h^{2+α}(𝕋^{n}), h^{1+α}(𝕋^{n})) .  (2.14)

In particular, the resolvent set of A is not empty. Moreover, the embedding h^{2+α}(𝕋^{n}) ↪↪ h^{1+α}(𝕋^{n}) is compact. Thus, A is an operator with compact resolvent and therefore ([9] Theorem III.8.29) its spectrum consists only of eigenvalues having finite multiplicity. Because of e_{k} ∈ h^{2+α}(𝕋^{n}) for all k ∈ ℤ^{n}, these functions are eigenfunctions of A corresponding to the eigenvalues λ_{k}. On the other hand, due to the embedding h^{2+α}(𝕋^{n}) ↪ H^{1}(𝕋^{n}), all eigenfunctions of A are also eigenfunctions of A_{L^{2}}. This implies the assertion.

The only obstacle for a proof of exponential stability of g = 0 in (2.9) is the fact that 0 ∈ σ(A). However, the eigenspace E_{0} corresponding to this eigenvalue consists precisely of the constants, and by Lemma 2.2 it is sufficient to consider the restriction Ã of A to h̃^{1+α}(𝕋^{n}) which is a complement of E_{0} in h^{1+α}. The following lemma provides the necessary results on Ã. To begin, observe that h̃^{2+α}(𝕋^{n}) is dense in h̃^{1+α}(𝕋^{n}).

Lemma 2.5 (The linearized operator on h̃^{1+α}(𝕋^{n}))

Consider Ã as an unbounded, densely defined operator on h̃^{1+α}(𝕋^{n}) with D(Ã) = h̃^{2+α}(𝕋^{n}). Then

Ã ∈ H (h̃^{2+α}(𝕋^{n}), h̃^{1+α}(𝕋^{n}))  (2.15)

and

σ(Ã) = {λ_{k} | k ∈ ℤ^{n}\{0}}.

Proof: Using the direct decomposition

h^{k+α}(𝕋^{n}) = h̃^{k+α}(𝕋^{n}) ⊕ E_{0}, k = 1, 2,

and the fact that AP ⊃ P A we can interpret A as a matrix operator

A = [ \begin{matrix}Ã & 0 \\ 0 & 0 \end{matrix} ]

and conclude (2.15) from [1], Corollary 1.6.3. For k ∈ ℤ^{n}\{0}, one has e_{k} ∈ h̃^{2+α}(𝕋^{n}) and hence λ_{k} ∈ σ(Ã). On the other hand, given λ ∈ ρ(A) and f ∈ h̃^{1+α}(𝕋^{n}), the equation

(A − λI)u = f  (2.16)

is uniquely solvable in h^{2+α}(𝕋^{n}). Applying P to both sides yields

P (A − λI)u = (A − λI)P u = P f = f,

i.e. P u solves (2.16), hence u = P u ∈ h̃^{2+α}(𝕋^{n}). Thus λ ∈ ρ(Ã), and consequently σ(Ã) ⊂ σ(A).

It remains to show that 0 ∈ ρ(Ã). Recall that the equation Ãu = f is equivalent to the existence of a v ∈ h^{2+α}(𝕋^{n} × (0, c)) such that

\begin{matrix}Δv & = & 0 & in 𝕋^{n} × (0, c), \\ \frac{∂v}{∂n} & = & f & on 𝕋^{n} × \{c\}, \\ \frac{∂v}{∂n} & = & 0 & on 𝕋^{n} × \{0\}, \\ v & = & u & on 𝕋^{n} × \{c\}. \end{matrix} \}  (2.17)

If f = 0 then it follows from the maximum principle that any v ∈ h^{2+α}(𝕋^{n} × (0, c)) satisfying (2.17)_{1}– (2.17)_{3} is a constant, hence u is a constant and thus u = 0 since u ∈ h̃^{2+α}(𝕋^{n}). This proves the injectivity of Ã. To show surjectivity, we fix f ∈ h̃^{1+α}(𝕋^{n}) and approximate f in the BUC^{1+α}-norm by smooth f^{j}, j ∈ ℕ. We have f^{j} = ∑_{k∈ℤ^{n}\{0}} a_{k}^{j}e_{k} where the a_{k}^{j} are rapidly decreasing in k for any fixed j. Setting

v^{j}(θ, y) := ∑_{k∈ℤ^{n}\\{0\}} \frac{a_{k}^{j}}{|k| sinh c|k|}e_{k}(θ) cosh |k|y  (2.18)

and u^{j} := v^{j}|_{𝕋^{n}×{c}} we find by standard arguments that u^{j} ∈ C̃^{∞}(𝕋^{n}) and

\begin{matrix}Δv^{j} & = & 0 & in 𝕋^{n} × (0, c), \\ \frac{∂v^{j}}{∂n} & = & f^{j} & on 𝕋^{n} × \{c\}, \\ \frac{∂v^{j}}{∂n} & = & 0 & on 𝕋^{n} × \{0\}, \\ v^{j} & = & u^{j} & on 𝕋^{n} × \{c\}, \end{matrix} \}  (2.19)

hence Ãu^{j} = f^{j}. For j, l ∈ ℕ, the well-known Schauder estimate for solutions of (2.19)_{1}– (2.19)_{3} yields

∥v^{j} − v^{l}∥_{BUC^{2+α}(𝕋^{n}×(0,c))} ≤ C (∥f^{j} − f^{l}∥_{BUC^{1+α}(𝕋^{n})} + ∥v^{j} − v^{l}∥_{L^{2}(𝕋^{n}×(0,c))}) .

(Usually, in the last term on the right the BUC^{0}(𝕋^{n} × (0, c))-norm is used. However, it follows from [14], Theorem 12.12, that both formulations are equivalent.) Clearly, one has

∥u^{j} − u^{l}∥_{BUC^{2+α}(𝕋^{n})} ≤ C∥v^{j} − v^{l}∥_{BUC^{2+α}(𝕋^{n}×(0,c))}.

Moreover, the obvious estimate

∥v^{j} − v^{l}∥_{L^{2}(𝕋^{n}×(0,c))} ≤ C∥f^{j} − f^{l}∥_{L^{2}(𝕋^{n})} ≤ C∥f^{j} − f^{l}∥_{BUC^{1+α}(𝕋^{n})}

shows that {v^{j}} and {u^{j}} are Cauchy sequences in h^{2+α}(𝕋^{n} × (0, c)) and h̃^{2+α}(𝕋^{n}), converging to some v ∈ h^{2+α}(𝕋^{n} × (0, c)) and u ∈ h̃^{2+α}(𝕋^{n}), respectively. Passage to the limit in (2.19) yields Ãu = f.

We are ready now for the proof of the main result of this section:

Theorem 2.6 (Exponential stability)

For any ω < λ_{1} = tanh c, there are positive constants r and C such that for any g_{0} ∈ O with ∥g_{0}∥_{BUC^{2+α}(𝕋^{n})} < r we have t^{+}(g_{0}) = ∞ and for the solution g of (2.9) the estimate

∥g(t)∥_{BUC^{2+α}(𝕋^{n})} + ∥g′(t)∥_{BUC^{1+α}(𝕋^{n})} ≤ Ce^{−ωt}∥g_{0}∥_{BUC^{2+α}(𝕋^{n})} ∀t ≥ 0

holds.

Proof: Setting X := h̃^{1+α}(𝕋^{n}), D := h̃^{2+α}(𝕋^{n}), G := Ã − Φ, ω_{0} := − sup{Re λ | λ ∈ σ(−Ã)} = λ_{1}, we find that all assumptions of Theorem 9.1.2. in [12] are satisfied because of (2.11), (2.12), and Lemmas 2.2 and 2.5. Our result follows from this theorem.

3 Flows in compressible porous media

In the description of the problem as well as in the notation, we will follow [6] in this section.

We consider a free boundary flow problem in a deformable porous medium. The geometric setting is the same as before, i.e. we assume that for fixed time the liquid occupies the domain

Ω̃_{f} := {(x, y) ∈ ℝ^{n} × ℝ | 0 < y < f(x)} ,

where f ∈ C^{1}(ℝ^{n}) is a uniformly positive function whose graph

Γ̃_{f} := {(x, y) ∈ ℝ^{n} × ℝ | y = f(x)}

represents the interface between the saturated and the dry part of the porous medium. Accordingly, we define

Ã:= {f ∈ C^{1}(ℝ^{n}) | inf_{x∈ℝ^{n}} f(x) > 0}

and for f ∈ C^{1}([0, T], Ã)

Ω̃_{f,T} := {(t, z) ∈ [0, T] × ℝ^{n+1} | z ∈ Ω̃_{f(t)}} .

Note that Ω̃_{f,T} is an unbounded striplike domain in [0, T]×ℝ^{n+1}. We are looking for a pair (u, f) of (sufficiently smooth) functions u : Ω̃_{f,T} ⟶ ℝ, f : [0, T] ⟶ Ã satisfying the system of equations

\begin{matrix}∂_{t}u − Δu & = & 0 & in Ω̃_{f(t)}, & t > 0, \\ ∂_{n+1}u & = & 0 & on Γ̃_{0}, & t ≥ 0, \\ u & = & f & on Γ̃_{f(t)}, & t ≥ 0, \\ ∂_{t}f + \sqrt{1 + |∇_{n}f|^{2}}∂_{ν}u & = & 0 & on Γ̃_{f(t)}, & t > 0, \\ u(0, ·) & = & u_{0} & in Ω̃_{f_{0}}, \\ f(0) & = & f_{0} & on ℝ^{n} \end{matrix} \}  (3.1)

with given initial data (u_{0}, f_{0}). As in the previous section, ∂_{ν}u denotes the derivative of u(t, ·) with respect to the outer unit normal on Γ̃_{f(t)} and the Laplacian involves the spatial coordinates only.

The crucial difference of (3.1) to the problem (2.1),(2.2) is the fact that the governing equation (3.1)_{1} is parabolic, hence it is not possible here to determine u(t, ·) for given f(t) at the fixed time t ∈ [0, T]. Instead, one has to consider a pair of coupled Cauchy problems for u and f. The system (3.1) occurs as a model for the gravity-driven motion of a compressible Newtonian liquid in a linearly elastic porous medium, where u represents the velocity potential of the flow. For the derivation of this model we refer to [6] and the references therein. Note, moreover, that (3.1) can also be interpreted as a one-phase Stefan problem with mixed boundary conditions.

As in the incompressible case, we demand spatial periodicity of f and u, i.e. (2.3) and (2.4), and consider (3.1) with Ã, Ω̃_{f(t)}, Γ̃_{f(t)}, Ω̃_{f,T} replaced by

A := {f ∈ C^{1}(𝕋^{n}) | min_{x∈𝕋^{n}} f(x) > 0} ,
Ω_{f(t)} := {(x, y) ∈ 𝕋^{n} × ℝ | 0 < y < f(t, x)} ,
Γ_{f(t)} := {(x, y) ∈ 𝕋^{n} × ℝ | y = f(t, x)} ,
Ω_{f,T} := {(t, z) ∈ [0, T] × 𝕋^{n} × ℝ | z ∈ Ω_{f(t)}} .

Let us transform (3.1) to a problem on the fixed reference domain Ω := 𝕋^{n} × (0, 1) using a diffeomorphism depending on the unknown function f. Following [6], we fix p > n + 2, s ∈ (0, \frac{1}{p}), α ∈ [\frac{1}{p}, 1 − \frac{n+1}{p}]. Furthermore, we write Γ_{i} := 𝕋^{n} × \{i\}, i = 0, 1, and denote by γ_{i} the trace operator from Ω to the boundary component Γ_{i}. Throughout this section we write W_{p}^{t}(Ω) and W_{p}^{t}(Γ_{0}) for the Sobolev-Slobodeckii spaces over Ω and Γ_{0}, respectively.

It is shown in [6], Section 2, that there is an extension operator

ρ_{0} ∈ L (W_{p}^{k+s− \frac{1}{p}} (Γ_{0}), W_{p}^{k+s}(Ω)) , k = 1, 2,

such that γ_{0}ρ_{0} is the identity on W_{p}^{k+s− \frac{1}{p}} (Γ_{0}) and ρ_{0}g vanishes near Γ_{1} for all g ∈

W_{p}^{1+s− \frac{1}{p}} (Γ_{0}). (Here and in the following, references to results in [6] are to be understood with the necessary changes due to our periodicity assumptions, i.e. with ℝ^{n} replaced by 𝕋^{n}. The results remain valid, and there are no essential new difficulties in the proofs.)

We set

Q_{0} := ∫_{Ω_{f_{0}}} u_{0} dx − \frac{1}{2} ∫_{𝕋^{n}} f_{0}^{2} dx′ + ∫_{𝕋^{n}} f_{0} dx′.  (3.2)

It will be shown later that for our purposes it is natural to demand Q_{0} > 0. Making this assumption, we define d > 0 as the (unique) solution of

\frac{(2π)^{n}}{2} d(d + 2) = Q_{0}  (3.3)

(see the remark after Lemma 3.3 below). Moreover, with

D := \{g ∈ W_{p}^{2+s− \frac{1}{p}} (Γ_{0}) | inf_{(x,y)∈Ω}(d + ρ_{0}g) > 0,

sup_{(x,y)∈Ω} \sqrt{|∇_{x}(ρ_{0}g)(x, y)|^{2} + |(1 − y)∂_{n+1}(ρ_{0}g)(x, y) − (ρ_{0}g)(x, y)|^{2}} < (1 ∧ δ)\}

and δ ∈ (0, d) fixed, we define for any g ∈ D the mapping θ_{g} : Ω ⟶ Ω_{g+d} by

θ_{g}(x, y) := (x, (1 − y)(d + (ρ_{0}g)(x, y)).

The corresponding pull-back and push-forward operators will be denoted by θ_{g}^{∗} and θ_{∗}^{g}, respectively. It is shown in [6], Lemma 2.2.a,b and Corollary 2.3, that D is an open

neighborhood of 0 in W_{p}^{2+s− \frac{1}{p}} (Γ_{0}), and θ_{g} ∈ Diff^{1+α}(Ω, Ω_{g+d}) with

θ_{∗}^{g} ∈ L (W_{p}^{k+s}(Ω), W_{p}^{k+s}(Ω_{g+d})) , k = 0, 1, 2,

for any g ∈ D.

This justifies the following approach: Using the obvious identification of Γ_{0} and 𝕋^{n} we demand f(t, ·) − d ∈ D, u(t, ·) ∈ W_{p}^{2+s}(Ω_{f(t)}) for t ∈ [0, T] and define functions g on [0, T] × Γ_{0} and v on [0, T] × Ω by

g := f − d, v(t, ·) := θ_{g(t)}^{∗}(u(t, ·) − d).

The problem (3.1) for (u, f) on the unknown domain Ω_{f,T} can be transformed now to a problem for (v, g) on the fixed cylindrical domain [0, T] × Ω, namely,

\begin{matrix}∂_{t}v + A(g)v + R(v, g) & = & 0 & in (0, T] × Ω, \\ v & = & g & on [0, T] × Γ_{0}, \\ B_{1}(g)v & = & 0 & on [0, T] × Γ_{1}, \\ ∂_{t}g + B_{0}(g)v & = & 0 & on (0, T] × Γ_{0}, \\ v(0, ·) & = & v_{0} & in Ω, \\ g(0, ·) & = & g_{0} & on Γ_{0}, \end{matrix} \}  (3.4)

where g_{0} := f_{0} − d ∈ D and v_{0} := θ_{g_{0}}^{∗}(u_{0} − d) ∈ W_{p}^{2+s}(Ω) are the given initial data,

A(g)v := −θ_{g}^{∗}Δ(θ_{∗}^{g}v),
B_{0}(g) := θ_{g}^{∗}(γ_{g+d}∇(θ_{∗}^{g}v) · n_{0}),
B_{1}(g) := θ_{g}^{∗}(γ_{0}∇(θ_{∗}^{g}v) · n_{1}),
R(v, g) := \frac{(y − 1)∂_{n+1}v}{G(g)} ρ_{0}(B_{0}(g)v),
G(g) := d + ρ_{0}g + (y − 1)∂_{n+1}(ρ_{0}g),

n_{0} := (−∇_{x}g, 1) and n_{1} := (0, −1) denote outer normals on Γ_{f} and Γ_{0}, respectively, and γ_{g+d} is the trace operator from Ω_{g+d} to Γ_{g+d}. Note that g ∈ D implies that G(g) has a positive lower bound on Ω and

G(g) = | det Dθ_{g}|.  (3.5)

For an explicit coordinate representation of the transformed operators A(g) and B_{i}(g) we refer to [6], Lemma 2.5. By transformation to Ω_{g+d} and Gauss’ theorem one has

− ∫_{Ω} A(g)v| det Dθ_{g}| dx = ∫_{Ω_{g+d}} Δ(θ_{∗}^{g}v) dx = ∫_{∂Ω_{g+d}} ∂_{ν}(θ_{∗}^{g}v) dσ

= ∫_{Γ_{g+d}} \frac{γ_{g+d}∇(θ_{∗}^{g}v) · n_{0}}{\sqrt{1 + |∇_{x}g|^{2}}} dσ + ∫_{Γ_{0}} γ_{0}∇(θ_{∗}^{g}v) · n_{1} dx′
= ∫_{Γ_{0}} B_{0}(g)v dx′ + ∫_{Γ_{1}} B_{1}(g)v dx′  (3.6)

for v ∈ W_{p}^{2+s}(Ω), g ∈ D.

In order to discuss (3.4) in a suitable abstract framework we set

E_{0} := W_{p}^{s}(Ω) × W_{p}^{1+s− \frac{1}{p}} (Γ_{0}),
E_{1} := \{(v, g) ∈ W_{p}^{2+s}(Ω) × W_{p}^{2+s− \frac{1}{p}} (Γ_{0}) | γ_{0}v = g, γ_{1}∂_{n+1}v = 0\} ,
D_{1} := {(v, g) ∈ E_{1} | g ∈ D}

and define Π : D_{1} ⟶ E_{0} by

Π(v, g) := (A(g)v + R(v, g), B_{0}(g)v).

Following [6], one finds that D_{1} is open in E_{1} and Π ∈ C^{ω}(D_{1}, E_{0}). With these definitions, we reformulate (3.4) as an abstract Cauchy problem

\begin{matrix}∂_{t}z + Π(z) & = & 0, \\ z(0) & = & z_{0} := (v_{0}, g_{0}) \end{matrix} \}  (3.7)

in E_{0} and it is clear that any solution z ∈ C([0, T], D_{1}) ∩ C^{1}([0, T], E_{0}) yields a solution of (3.4) via z = (v, g).

Moreover, for ρ > 0, σ > 0 we define

W_{ρ,σ} := \{(v, g) ∈ D_{1} | ∥v∥_{W_{p}^{2+s}(Ω)} < ρ, ∥∂_{n+1}v∥_{C^{α}(Ω)} < σ, ∥γ_{0}∂_{n+1}v∥_{W_{p}^{1+s− \frac{1}{p}} (Γ_{0})} < σ \} .

It is easily seen that W_{ρ,σ} is an open neighborhood of 0 in D_{1}.

The crucial result on the parabolic character of (3.7) is the following ([6] Corollary 3.6):

Lemma 3.1 (Parabolicity of (3.7))

For any ρ > 0 there is a σ > 0 such that

∂Π(z) ∈ H(E_{1}, E_{0}) ∀z ∈ W_{ρ,σ}.

This is the crucial fact in the proof of the classical solvability of (3.7). More precisely, the following results are proved in [6], cf. Theorem 3.8.:

Theorem 3.2 (Solvability of (3.7) and time regularity of the solution)

Let ρ > 0 be given. There exists σ > 0 such that for each z_{0} ∈ W_{ρ,σ} there is a t^{+} := t^{+}(z_{0}) > 0 and a unique maximal solution

z ∈ C([0, t^{+}), W_{ρ,σ}) ∩ C^{1}([0, t^{+}), E_{0})

of (3.7). Moreover,

z ∈ C^{∞} ((0, t^{+}), W_{p}^{2+s̃}(Ω) × W_{p}^{2+s̃− \frac{1}{p}} (Γ_{0}))

for any s̃ ∈ (0, s).

It is easily checked that z = (0, 0) is a stationary solution of (3.7). As in the previous section, the first step in the investigation of its solvability is the identification of a conserved quantity.

Lemma 3.3 (A conserved quantity)

Let z = (v, g) be a solution of (3.7) in the sense of Theorem 3.2. Then the functional

I(z) := ∫_{Ω}(v + d)| det Dθ_{g}| dx − \frac{1}{2} ∫_{Γ_{0}}(g + d)^{2} dx′ + ∫_{Γ_{0}}(g + d) dx′

is constant on [0, t^{+}).

Proof: Due to the continuous embeddings E_{1} ↪ BUC^{1+α}(Ω) × BUC^{1+α}(Γ_{0}) and

W_{p}^{2+s̃}(Ω) × W_{p}^{2+s̃− \frac{1}{p}} (Γ_{0}) ↪ BUC^{1+α}(Ω) × BUC^{1+α}(Γ_{0}) we have

z ∈ C ([0, t^{+}), BUC^{1+α}(Ω) × BUC^{1+α}(Γ_{0}))∩C^{1} ((0, t^{+}), BUC^{1+α}(Ω) × BUC^{1+α}(Γ_{0})) ,

and the mapping t ↦ I(z(t)) is continuous on [0, t^{+}) and differentiable on (0, t^{+}) with derivative

İ(z) = ∫_{Ω} ∂_{t}v| det Dθ_{g}| dx + ∫_{Γ_{0}} ∫_{0}^{1}(v + d)∂_{y}[(y − 1)ρ_{0}(∂_{t}g)] dydx′

− ∫_{Γ_{0}} ∂_{t}g(g + d) dx′ + ∫_{Γ_{0}} ∂_{t}g dx′

= ∫_{Ω} ∂_{t}v| det Dθ_{g}| dx + ∫_{Γ_{0}} ∫_{0}^{1} ∂_{y}[(v + d)(y − 1)ρ_{0}(∂_{t}g)] dydx′

− ∫_{Γ_{0}} ∫_{0}^{1} \frac{∂_{y}v(y − 1)ρ_{0}(∂_{t}g)}{G(g)}| det Dθ_{g}| dydx′ − ∫_{Γ_{0}} ∂_{t}g(g + d) dx′ + ∫_{Γ_{0}} ∂_{t}g dx′

= ∫_{Ω}(∂_{t}v + R(v, g))| det Dθ_{g}| dx − ∫_{Γ_{0}}(g + d)B_{0}(g)v dx′

− ∫_{Γ_{0}} ∂_{t}g(g + d) dx′ + ∫_{Γ_{0}} ∂_{t}g dx′

= − ∫_{Ω} A(g)v| det Dθ_{g}| dx + ∫_{Γ_{0}} ∂_{t}g dx′
= ∫_{Γ_{0}} B_{0}(g)v dx′ + ∫_{Γ_{1}} B_{1}(g)v dx′ + ∫_{Γ_{0}} ∂_{t}g dx′ = 0,

where (3.5) and (3.6) have been used.

Remark: It is natural to reinterpret Lemma 3.3 in terms of the original problem (3.1). For solutions of this problem, the quantity

Q(u, f) = ∫_{Ω_{f(t)}} u dx − \frac{1}{2} ∫_{𝕋^{n}} f^{2} dx + ∫_{𝕋^{n}} f dx

is conserved during the evolution. For positive Q(u, f), this fact determines to which of the possible equilibria (u, f) = (c, c), c > 0, a solution with given initial data (u_{0}, f_{0}) may tend. In view of (3.2), (3.3), and Q(c, c) = \frac{(2π)^{n}}{2} c(c + 2) we find c = d. On the other hand, for initial data with

Q(u_{0}, f_{0}) < 0  (3.8)

one obtains that a corresponding solution cannot tend to any equilibrium. An elementary calculation shows that (3.8) can hold only if ∂_{n+1}u_{0} is relatively large. This gives rise to the conjecture that solutions of (3.1) with (3.8) will cease to exist at a finite time at which the strict positivity of f is lost, i.e. at which the upper boundary component of the liquid domain will touch the bottom. (Of course, it is by no means clear whether there are further blowup mechanisms.) Note, moreover, that if Q_{0} > 0 then for any ρ > 0 there is a σ > 0 such that I(v, g) > 0 for all (v, g) ∈ W_{ρ,σ}.

In a next step, we derive the necessary information on the spectrum of the operator A := ∂Π(0). Due to the dense embedding E_{1} ↪ E_{0}, A can and will be considered as an unbounded, densely defined operator on E_{0} with D(A) = E_{1}. Lemma 3.3 in [6] applied to z = (v, g) = 0 shows that

A(w, h) = (A(0)w, B_{0}(0)w).

Lemma 3.4 (The linearized operator)

The spectrum of A consists only of nonnegative real eigenvalues.

Proof: From Lemma 3.1 we know that A ∈ H(E_{1}, E_{0}). As in the proof of Lemma 2.4, it follows from this together with the compactness of the embedding E_{1} ↪ E_{0} that σ(A) consists only of eigenvalues. On E_{1} we define the scalar product (·|·) by

((v, g)|(w, h)) := d ∫_{Ω} vw dx + ∫_{Γ_{0}} gh dx′ = ∫_{Ω_{d}} θ_{∗}^{0}vθ_{∗}^{0}w dx + ∫_{Γ_{d}} θ_{∗}^{0}gθ_{∗}^{0}h dx′.

The operator A is symmetric and positive semidefinite with respect to this scalar product. Indeed, for arbitrary (v, g), (w, h) ∈ E_{1},

(A(v, g)|(w, h)) = − ∫_{Ω_{d}} Δ(θ_{∗}^{0}v)θ_{∗}^{0}w dx + ∫_{Γ_{d}} ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}h dx′
= − ∫_{Ω_{d}} θ_{∗}^{0}vΔ(θ_{∗}^{0}w) dx + ∫_{Γ_{d}} (θ_{∗}^{0}v∂_{ν}(θ_{∗}^{0}w) − ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}w) dx′
+ ∫_{Γ_{0}} (θ_{∗}^{0}v∂_{ν}(θ_{∗}^{0}w) − ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}w) dx′ + ∫_{Γ_{d}} ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}h dx′
= − ∫_{Ω_{d}} θ_{∗}^{0}vΔ(θ_{∗}^{0}w) dx + ∫_{Γ_{d}} θ_{∗}^{0}g∂_{ν}(θ_{∗}^{0}w) dx′ = ((v, g)|A(w, h))

and

(A(v, g)|(v, g)) = − ∫_{Ω_{d}} Δ(θ_{∗}^{0}v)θ_{∗}^{0}v dx + ∫_{Γ_{d}} ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}g dx′
= ∫_{Ω_{d}} |∇(θ_{∗}^{0}v)|^{2} dx − ∫_{Γ_{d}} ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}v dx′ − ∫_{Γ_{0}} ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}v dx′ + ∫_{Γ_{d}} ∂_{ν}(θ_{∗}^{0}v)θ_{∗}^{0}g dx′
= ∫_{Ω_{d}} |∇(θ_{∗}^{0}v)|^{2} dx ≥ 0,

where we have used that γ_{0}v = g, γ_{0}w = h, γ_{1}∂_{n+1}v = γ_{1}∂_{n+1}w = 0.

On the complexification of E_{1} we introduce the scalar product [·|·] by

[z_{1}|z_{2}] = (x_{1}|x_{2}) + (y_{1}|y_{2}) + i((y_{1}|x_{2}) − (y_{2}|x_{1})),

where z_{j} = x_{j} + iy_{j}, x_{j}, y_{j} ∈ E_{1}, j = 1, 2. Suppose now λ is an eigenvalue of A with eigenfunction z ≠ 0. This implies

λ[z|z] = [Az|z] = [z|Az] = \overline{λ}[z|z],

hence \overline{λ} = λ ∈ ℝ, and z can and will be chosen to be real. Then (Az|z) = λ(z|z) ≥ 0, hence λ ≥ 0.

As in the incompressible case discussed above, it is not hard to see that 0 is an eigenvalue of A. Therefore, by exploiting the conservation of I, one again has to consider a reduced system in which this eigenvalue does not occur. In the present situation, however, the evolution does not proceed in an invariant subspace but, according to Lemma 3.3, in the invariant subset

M′ := {z ∈ E_{1} | I(z) = Q_{0}}.

Using (3.5), it is straightforward to calculate the Fréchet derivative of I ∈ C^{ω}(D_{1}, ℝ) at (v, g) ∈ D_{1} as

∂I(v, g)[w, h] = ∫_{Ω} w| det Dθ_{g}| dx + ∫_{Γ_{0}} ∫_{0}^{1}(v + d)∂_{y}[(y − 1)ρ_{0}h] dydx′
+ ∫_{Γ_{0}} h(1 − (g + d)) dx′,

(w, h) ∈ E_{1}. This linear functional is nondegenerate, and it follows that M := M′ ∩D_{1} is a smooth hypersurface in E_{1}. Note that 0 is the only stationary point of (3.7) in M. Our strategy is to describe the restriction of the dynamical system given by (3.7) to M using a local diffeomorphism mapping M near 0 onto its tangent space Ẽ_{1} := T_{0}M = ker ∂I(0, 0) where

∂I(0, 0) = d ∫_{Ω} w dx + d ∫_{Γ_{0}} ∫_{0}^{1} ∂_{y}[(y − 1)ρ_{0}h] dydx′ + (1 − d) ∫_{Γ_{0}} h dx′
= d ∫_{Ω} w dx + ∫_{Γ_{0}} h dx′.

Let us define the projection P ∈ L(E_{0}) ∩ L(E_{1}) by

P (v, g) := (v, g) − \frac{(1, 1)}{(2π)^{n}(d + 1)}((v, g)|(1, 1)).

Clearly, P (E_{1}) = Ẽ_{1} and ker P = {(c, c) | c ∈ ℝ}. Defining Ẽ_{0} := P (E_{0}), we obtain the direct decomposition

E_{i} = Ẽ_{i} ⊕ ker P, i = 0, 1.

The operator A vanishes on ker P, hence A = AP. Furthermore,

(I − P)A(w, h) = \frac{(1, 1)}{(2π)^{n}(d + 1)}(A(w, h)|(1, 1)) = \frac{(1, 1)}{(2π)^{n}(d + 1)}((w, h)|A(1, 1)) = 0

for (w, h) ∈ E_{1}, hence A = AP = P A on E_{1}. We define the restriction Ã := A|Ẽ_{1} and collect the properties of Ã that will be important for our purpose.

Lemma 3.5 (The linearized operator on the restricted space)

Ã ∈ H(Ẽ_{1}, Ẽ_{0}) and the spectrum of Ã consists only of positive eigenvalues.

Proof: By the same arguments as in the proof of Lemma 2.5, we obtain Ã ∈ H(Ẽ_{1}, Ẽ_{0}) and σ(Ã) ⊂ σ(A). It remains to show that Ã ∈ Isom(Ẽ_{1}, Ẽ_{0}). This will be done by discussing the Neumann BVP

\begin{matrix}−Δu & = & ϕ & in Ω_{d}, \\ ∂_{ν}u & = & ψ & on Γ_{d}, \\ ∂_{ν}u & = & 0 & on Γ_{0}, \\ ∫_{Ω_{d}} u dx + ∫_{Γ_{d}} u dx′ & = & 0. \end{matrix} \}  (3.9)

We will show that that for any (ϕ, ψ) ∈ W_{p}^{s}(Ω_{d}) × W_{p}^{s+1− \frac{1}{p}} (Γ_{d}) satisfying

∫_{Ω_{d}} ϕ dx + ∫_{Γ_{d}} ψ dx′ = 0  (3.10)

there exists precisely one solution of (3.9) in W_{p}^{s+2}(Ω_{d}). To show uniqueness, we set (ϕ, ψ) = 0 and obtain from (3.9)_{1}– (3.9)_{3} that ∫_{Ω_{d}} |∇u|^{2} dx = 0, hence u is constant, and it follows from (3.9)_{4} that u = 0. To show existence, we first remark that (ϕ, ψ) ∈ L^{p}(Ω_{d}) × L^{p}(Γ_{d}) ↪ L^{2}(Ω_{d}) × L^{2}(Γ_{d}), and a discussion of the weak formulation shows that (3.9) has a unique solution u ∈ H^{1}(Ω_{d}) satisfying an estimate

∥u∥_{L^{2}(Ω_{d})} ≤ C (∥ϕ∥_{L^{2}(Ω_{d})} + ∥ψ∥_{L^{2}(Γ_{d})}) ≤ C (∥ϕ∥_{W_{p}^{s}(Ω_{d})} + ∥ψ∥_{W_{p}^{s+1− \frac{1}{p}} (Γ_{d})}) . (3.11)

Consider the case where ϕ and ψ are smooth. In this case, it follows from standard elliptic regularity results that u is smooth, and the a priori estimate given in [13] Theorem 5.3.4 together with Theorem 12.12 in [14] and (3.11) yields

∥u∥_{W_{p}^{s+2}(Ω_{d})} ≤ C (∥ϕ∥_{W_{p}^{s}(Ω_{d})} + ∥ψ∥_{W_{p}^{s+1− \frac{1}{p}} (Γ_{d})} + ∥u∥_{L^{2}(Ω_{d})})
≤ C′ (∥ϕ∥_{W_{p}^{s}(Ω_{d})} + ∥ψ∥_{W_{p}^{s+1− \frac{1}{p}} (Γ_{d})}) .

The solvability of (3.9) in W_{p}^{s+2}(Ω_{d}) in the general case follows now by a standard approximation argument.

To complete the proof, we set (ϕ, ψ) := (θ_{∗}^{0}w, θ_{∗}^{0}h) for (w, h) ∈ Ẽ_{0}. Then (3.10) is satisfied, and A(v, g) = (w, h) with (v, g) ∈ Ẽ_{1} if and only if v = θ_{0}^{∗}u, g = γ_{0}v, i.e. this operator equation is always uniquely solvable.

Lemma 3.6 (A local chart for M)

There are neighborhoods U_{1} and V_{1} of 0 in Ẽ_{1} and M, respectively, and a diffeomorphism Φ ∈ Diff^{∞}(U_{1}, V_{1}) such that Φ ∘ P = id_{V_{1}} and ∂Φ(0) = idẼ_{1}.

Proof: Define the function F ∈ C^{∞}(Ẽ_{1} × E_{1}, Ẽ_{1} × ℝ) by

F (x, z) := (P z − x, I(z) − Q_{0}).

Clearly F (0, 0) = 0, the partial Fréchet derivatives in (0, 0) are given by

∂_{1}F (0, 0)[h] = (−h, 0),
∂_{2}F (0, 0)[k] = (P k, ∂I(0, 0)[k]),

and ∂_{2}F (0, 0) is invertible with

−(∂_{2}F (0, 0))^{−1}∂_{1}F (0, 0)[h] = h, h ∈ Ẽ_{1}.

By this, all assertions follow from the Implicit Function Theorem.

Let ρ > 0 be given and choose σ > 0 according to Theorem 3.2. Furthermore, we choose open neighborhoods V ⊂ V_{1} and U ⊂ U_{1} of 0 in M and Ẽ_{1}, respectively, such that \overline{V} ⊂ W_{ρ,σ} ∩ V_{1}, U = P [V] and Φ is Lipschitz continuous on U. Assume z_{0} ∈ V and define

t̃^{+} := sup{t ∈ [0, t^{+}) |z(t) ∈ V},
z̃(t) := P z(t),  t ∈ [0, t̃^{+}),
Π̃ := P ∘ Π ∘ Φ

with z and t^{+} from Theorem 3.2. Note that Π̃ ∈ C^{∞}(U_{1}, Ẽ_{0}), z̃ ∈ C([0, t̃^{+}), U) ∩ C^{1}([0, t̃^{+}), Ẽ_{0}) and due to

P Π(z(t)) = P Π(Φ(P z(t))) = Π̃(z̃(t))

it satisfies

\begin{matrix}∂_{t}z̃ + Π̃(z̃) & = & 0, \\ z̃(0) & = & P z_{0} \end{matrix} \}  (3.12)

on [0, t̃^{+}).

Theorem 3.7 (Exponential stability for the projected problem)

There are positive constants r, M, and ω such that for any z_{0} with ∥P z_{0}∥_{E_{1}} < r the following holds:

i) z̃ is the only solution of (3.12) in C([0, t̃^{+}), U) ∩ C^{1}([0, t̃^{+}), Ẽ_{0}),

ii) z̃ can be uniquely extended to a solution of (3.12) in C([0, ∞), U) ∩ C^{1}([0, ∞), Ẽ_{0}), still denoted by z̃,

iii) z̃ satisfies the estimate

∥z̃(t)∥_{E_{1}} + ∥∂_{t}z̃(t)∥_{E_{0}} ≤ Me^{−ωt}∥P z_{0}∥_{E_{1}}.

Proof: Using Lemma 3.6 we find ∂Π̃(0) = P ∂Π(0) = Ã. As σ(A) is closed, we obtain from Lemma 3.5 that

ω_{0} := inf{Re λ | λ ∈ σ(Ã)} > 0.

The results follow immediately from this and Ã ∈ H(Ẽ_{1}, Ẽ_{0}) by [12], Theorem 9.1.2.

Corollary 3.8 (Exponential stability for the original problem)

There are positive constants r, M, and ω such that for any z_{0} with ∥z_{0}∥_{E_{1}} < r one has t^{+} = ∞ in Theorem 3.2 and the estimate

∥z(t)∥_{E_{1}} ≤ Me^{−ωt}∥z_{0}∥_{E_{1}}, t ≥ 0,

holds.

Proof: Let r := \frac{r̃}{∥P ∥_{L(E_{1})}} where r̃ denotes the constant r from Theorem 3.7 and assume

∥z_{0}∥_{E_{1}} < r. This implies the statements of Theorem 3.7, in particular, t̃^{+} = t^{+} and thus

{z(t) | t ∈ [0, t^{+})} = Φ[{z̃(t) | t ∈ [0, t^{+})}],

i.e. the trajectory of the maximal solution of (3.7) is the image of a relatively compact set under a continuous map, hence it is relatively compact. By [12], Proposition 8.2.1 and Lemma 8.2.2., this implies lim_{t→t^{+}} z(t) ∈ ∂W_{ρ,σ} or t^{+} = ∞. The first possibility is excluded because \overline{V} ⊂ W_{ρ,σ} and W_{ρ,σ} is open, hence t^{+} = ∞. The estimate of the corollary is an immediate consequence of Theorem 3.7 and the Lipschitz continuity of Φ and P.

Due to the smoothness of z in t, the stability results can also be translated back to the original problem (3.1).

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