The Inviscid Limit to a Contact Discontinuity for the Compressible Navier--Stokes--Fourier System Using the Relative Entropy Method

Abstract

We consider the zero heat conductivity limit to a contact discontinuity for the mono-dimensional full compressible Navier--Stokes--Fourier system. The method is based on the relative entropy method and does not assume any smallness conditions on the discontinuity nor on the bounded variation norm of the initial data. It is proved that for any viscosity $\nu\geq0$, the solution of the compressible Navier--Stokes--Fourier system (with well-prepared initial value) converges, when the heat conductivity $\kappa$ tends to zero, to the contact discontinuity solution to the corresponding Euler system. We obtain the decay rate $\kappa^{\frac{1}{2}}$. It implies that the heat conductivity dominates the dissipation in the regime of the limit to a contact discontinuity. This is the first result, based on the relative entropy, of an asymptotic limit to a discontinuous solution for a system.

Keywords

  1. contact discontinuity
  2. inviscid limit
  3. compressible Navier--Stokes--Fourier system
  4. relative entropy method

MSC codes

  1. 35Q30
  2. 76D05
  3. 35L65

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Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4350 - 4359
ISSN (online): 1095-7154

History

Submitted: 28 May 2015
Accepted: 15 September 2015
Published online: 10 November 2015

Keywords

  1. contact discontinuity
  2. inviscid limit
  3. compressible Navier--Stokes--Fourier system
  4. relative entropy method

MSC codes

  1. 35Q30
  2. 76D05
  3. 35L65

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