From the highly compressible Navier-Stokes equations to the Porous Medium equation – rate of convergence
Résumé
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 1 √ ε for ε going to 0. When the initial velocity is related to the gradient of the initial density, a solution to the continuity equation– ρ ε converges to the unique solution to the porous medium equation [13, 14]. For viscosity coefficient µ(ρ ε) = ρ α ε with α > 1, we obtain a rate of convergence of ρ ε in L ∞ (0, T ; H −1 (R)); for 1 < α ≤ 3 2 the solution ρ ε converges in L ∞ (0, T ; L 2 (R)). For compactly supported initial data, we prove that most of the mass corresponding to solution ρ ε is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of ε.
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