The slightly compressible Brinkman-Forcheimer equations and its incompressible approximation

Abstract

This paper investigates the slightly compressible Brinkman–Forchheimer equations (BFEs): ∂tu−Δxu+∇xp+f(u)=g with D−1(t)∂tp+divu=0 in a bounded 3D domain with Dirichlet boundary conditions. The features of this problem is that, formally, this system is partially dissipative, and will recover to incompressible BFEs when the time-dependent coefficient D(t) goes to infinity as t→∞. The well-posedness and dissipation are obtained in the natural energy space. In addition, our result reveals that the velocity field u can be approximated by the solution uin of incompressible BFEs in L2 provided that ∂tpin is uniform bounded in L2, here pin is the pressure of incompressible BFEs.