Global existence and large time behavior for the system of compressible adiabatic flow through porous media in $\mathbb{R}^{3}$

TL;DR

This paper proves the global existence and describes the large-time decay behavior of classical solutions for compressible adiabatic flow through porous media in three dimensions, highlighting faster velocity decay compared to Navier-Stokes equations without heat conductivity.

Contribution

It establishes the global existence and uniqueness of solutions near equilibrium and analyzes their decay rates, extending understanding of compressible flow in porous media.

Findings

Pressure converges to equilibrium at rate (1+t)^{-3/4}

Velocity decays faster at rate (1+t)^{-5/4}

Results extend decay analysis to compressible adiabatic flow in porous media

Abstract

The system of compressible adiabatic flow through porous media is considered in $\mathbb{R}^{3}$ in the present paper. The global existence and uniqueness of classical solutions are obtained when the initial data is near its equilibrium. We also show that the pressure of the system converges to its equilibrium state at the same $L^2$-rate $(1+t)^{-3/4}$ as the Navier-Stokes equations without heat conductivity, but the velocity of the system decays at the $L^2$-rate $(1+t)^{-5/4}$, which is faster than the $L^2$-rate $(1+t)^{-3/4}$ for the Navier-Stokes equations without heat conductivity [3].