Large time behavior for the system of compressible adiabatic flow through porous media in \(\mathbb{R}^3\). (English) Zbl 1416.35198
Summary: The Cauchy problem of the system of compressible adiabatic flow through porous media in \(\mathbb{R}^3\) 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 G. Wu et al. [J. Differ. Equations 255, No. 5, 865–880 (2013; Zbl 1404.35372)], 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.
MSC:
| 35Q31 | Euler equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 76S05 | Flows in porous media; filtration; seepage |
| 34B45 | Boundary value problems on graphs and networks for ordinary differential equations |
Keywords:
Euler equation with damping; global existence; large time behavior; high frequency and low frequency decompositionCitations:
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