Fluid dynamic limit to the Riemann Solutions of  Euler      equations: I. Superposition of rarefaction waves and contact discontinuity

Feimin Huang; Yi Wang; Tong Yang

doi:10.3934/krm.2010.3.685

Article Contents

2010, Volume 3, Issue 4: 685-728. Doi: 10.3934/krm.2010.3.685

This issue Previous Article $L^1$ averaging lemma for transport equations with Lipschitz force fields Next Article 1D Vlasov-Poisson equations with electron sheet initial data

Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity

1.
Institute of Applied Mathematics, AMSS and Hua Loo-Keng Key Laboratory of Mathematics, Academia Sinica, Beijing 100190
2.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received: July 31, 2010

Revised: September 30, 2010

Published: November 2010

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Abstract

Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters are also obtained. It is noted that this is the first rigorous proof of this limit for a Riemann solution with superposition of three waves even though the fluid dynamic limit for a single wave has been proved.

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Mathematics Subject Classification: Primary: 35Q30, 35Q20, 76N15, 76P05; Secondary: 35L65, 82B40, 82C40.

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