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.
Highlights
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 the first part of this paper, we investigate the fluid dynamic limit of the compressible Navier-Stokes equations when the corresponding Euler equations have the Riemann solution as a superposition of two rarefaction waves and a contact discontinuity
In the second part of the paper, we study the hydrodynamic limit of the Boltzmann equation [2] with slab symmetry ft + ξ1fx
Summary
With s denoting the entropy of the gas and A, R > 0 , γ > 1 being the gas parameters. How to justify the zero dissipation limit to the Euler equations with basic wave patterns is a natural and difficult problem. In the first part of this paper, we investigate the fluid dynamic limit of the compressible Navier-Stokes equations when the corresponding Euler equations have the Riemann solution as a superposition of two rarefaction waves and a contact discontinuity. How to verify the fluid limit from Boltzmann equation to the Euler equations with basic wave patterns becomes an natural problem. Xin-Zeng [31] proved the fluid dynamic limit of the compressible Navier-Stokes equations and Boltzmann equation to the Euler equations with non-interacting rarefaction waves. We will study the hydrodynamic limit of the Boltzmann equation when the corresponding Euler equations have a Riemann solution as a superposition of two rarefaction waves and a contact discontinuity.
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