Large time behavior for the system of compressible adiabatic flow through porous media in R3

Abstract

The Cauchy problem of the system of compressible adiabatic flow through porous media in R 3 is considered. We obtain the global existence and large time behavior of the solution when the initial data is close to its equilibrium in H 3 -norm. This is a continuing work of [31] , where due to non-dissipative property of the entropy s , an additional assumption that the initial data is bounded in L 1 -norm plays a key role in closing the a priori energy estimates and hence establishing the global existence of the solution. In this paper, to remove this assumption, we first decompose the solution U into high frequency part U h and low frequency part U l . Then by constructing delicate energy functionals for U h and optimal decay rates for U l , we can get the desired energy estimates to close the a priori assumption. Furthermore, we get the optimal L 2 − L 2 decay rate of the solution and show that the velocity decays faster than the density.