FROM THE HIGHLY COMPRESSIBLE NAVIER-STOKES EQUATIONS TO THE POROUS MEDIUM EQUATION RATE OF CONVERGENCE

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 epsilon(-1/2) for epsilon going to 0. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations-rho(epsilon) converge to the unique solution to the porous medium equation [14, 13]. For viscosity coefficient mu(rho(epsilon)) = rho(alpha)(epsilon) with alpha > 1, we obtain a rate of convergence of rho(epsilon) in L-infinity(0, T; H-1(R)); for 1 < alpha <= 3/2 the solution rho(epsilon) converges in L-infinity(0, T; L-2 (R)). For compactly supported initial data, we prove that most of the mass corresponding to solution p, is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of epsilon.