The asymptotic behavior (as well as the global existence) of classical solutions to the 3D compressible Euler equations are considered. For polytropic perfect gas , time asymptotically, it has been proved by Pan and Zhao (2009) that linear damping and slip boundary effect make the density satisfying the porous medium equation and the momentum obeying the classical Darcy's law. In this paper, we use a more general method and extend this result to the 3D compressible Euler equations with nonlinear damping and a more general pressure term. Comparing with linear damping, nonlinear damping can be ignored under small initial data.
"Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition." J. Appl. Math. 2012 1 - 16, 2012. https://doi.org/10.1155/2012/584680