The Research Summary Of The Compressible Euler Equations With Damping And Vacuum

We study the asymptotic behavior of the solutions to the compressible Euler equations with damping and vacuum. The previous works concern the case away from vacuum where the system is strictly hyperbolic. And for the vacuum case, they focus on the small smooth solutions or piecewise smooth Riemann solutions. In present paper, we offer the large-time asymptotic behavior for the L? entropy weak solution with vacuum, obtained by the compensated compactness theory. As time tend to infinity , the density obeys the well-know porous medium equation and the momentum is formulated by Darcy's law.