Résumé (EN)

We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to ϵ−1/2 for ϵ going to 0. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations --ρϵ converge to the unique solution to the porous medium equation [14,13]. For viscosity coefficient μ(ρϵ)=ραϵ with α>1, we obtain a rate of convergence of ρϵ in L∞(0,T;H−1(R)); for 1<α≤32 the solution ρϵ converges in L∞(0,T;L2(R)). For compactly supported initial data, we prove that most of the mass corresponding to solution ρϵ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of ϵ.