Uniform boundedness in time and large time behavior of weak solutions to a kind of dissipative system

This paper deals with the uniform boundedness (as well as the existence) and large time behavior of the weak entropy solutions to a kind of compressible Euler equation with dissipation effect. The existence and uniform boundedness in time of weak solutions are proved by using the Lax–Friedrichs scheme and compensate compactness. Time asymptotically, the density is showed to satisfy a kind of nonlinear Fokker–Planck equation and the momentum obeys to the Darcy’s law. As a by product, the exponentially decay rate is obtained.