From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence

Boris  Haspot; Ewelina  Zatorska

doi:10.3934/dcds.2016.36.3107

Article Contents

2016, Volume 36, Issue 6: 3107-3123. Doi: 10.3934/dcds.2016.36.3107

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From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence

Boris Haspot^1, and
Ewelina Zatorska^2,

1.
Ceremade UMR CNRS 7534 Universite Paris Dauphine, Place du Marechal DeLattre De Tassigny, 75775 PARIS CEDEX 16
2.
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw

Received: March 31, 2015

Revised: September 30, 2015

Published: May 2017

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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 $\epsilon^{-1/2}$ 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 $\mu(\rho_\epsilon)=\rho_\epsilon^\alpha$ with $\alpha>1$, we obtain a rate of convergence of $\rho_\epsilon$ in $L^\infty(0,T; H^{-1}(\mathbb{R}))$; for $1<\alpha\leq\frac{3}{2}$ 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 $\epsilon$.

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Mathematics Subject Classification: Primary: 35B25, 35Q30; Secondary: 35K55.

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