From the text: We study asymptotic behaviour of compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping
\[ \begin{cases} \rho_t+(\rho u)_x=0 ,\\ (\rho u)_t+(\rho u^2+p(\rho))_x=-\alpha \rho u ,\\ \rho(x,0)=\rho_0 (x),\\ u(x,0)=u_0(x) . \end{cases} \]
As \(t\to \infty\), the density is conjectured to obey the well-known porus medium equation and the momentum is expected to be formulated by Darcy’s law.
\[ \begin{cases} \overline{\rho}_t=(\overline{\rho} ^{\gamma})_{xx} ,\\ \overline m =-(\overline{\rho}^{\gamma})_x . \end{cases} \]
In this paper, we prove that any \(L^\infty\) weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges strongly in the natural \(L^1\) topology with decay rates to the Barenblatt profile of the porus medium equation. The density function tends to the Barenblatt solution of the porus medium equation, while the momentum is described by Darcy’s law. The results are achieved through a comprehensive entropy analysis, capturing the dissipative character of the problem.