On the Barotropic Compressible Navier–Stokes Equations

Abstract

We consider barotropic compressible Navier–Stokes equations with density dependent viscosity coefficients that vanish on vacuum. We prove the stability of weak solutions in periodic domain Ω = T N and in the whole space Ω = ℝ N , when N = 2 and N = 3. The pressure is given by p(ρ) = ργ and our result holds for any γ > 1. Note that our notion of weak solutions is not the usual one. In particular we require some regularity on the initial density (which may still vanish). On the other hand, the initial velocity must satisfy only minimal assumptions (a little more than finite energy). Existence results for such solutions can be obtained from this stability analysis.