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