From the highly compressible Navier-Stokes equations to the Porous Medium equation - rate of convergence

Abstract

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 $\frac{1}{\sqrt{\epsilon }}$ for $\epsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, a solution to the continuity equation-${\rho }_{\epsilon }$ converges to the unique solution to the porous medium equation [13,14]. For viscosity coefficient $\mu ({\rho }_{\epsilon })={\rho }_{\epsilon }^{\alpha }$ with $\alpha >1$, we obtain a rate of convergence of ${\rho }_{\epsilon }$ in $L^{\infty }(0,T;H^{-1}(\mathbb{R}))$; for $1<\alpha \le \frac{3}{2}$ the solution ${\rho }_{\epsilon }$ converges in $L^{\infty }(0,T;L^2(\mathbb{R}))$. For compactly supported initial data, we prove that most of the mass corresponding to solution ${\rho }_{\epsilon }$ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of $\epsilon$.

19 pages

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