Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping and Vacuum

We study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t →∞, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy’s law. In this paper, we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data. We prove that any L∞ weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges, strongly in Lp with decay rates, to matching Barenblatt’s profile of the porous medium equation. The density function tends to the Barenblatt’s solution of the porous medium equation while the momentum is described by Darcy’s law.