This is the abstract of the paper "Stability of the equilibria for spatially periodic flows in porous media" by J. Escher and G. Prokert. For the whole paper send the command get evolve-l 99-00051 to: "listserv@rz.uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. 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.