Initial boundary value problem for compressible Euler equations with damping

Abstract

We construct global L ∞ entropy weak solutions to the initial boundary value problem for the damped compressible Euler equations on bounded domains with physical boundaries. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on entropy principle, we show that the physical weak solution converges to steady states exponentially fast in time. We also prove that the same is true for the related initial boundary value problems of porous medium equation and thus justifies the validity of Darcy's law in large time.