Fluid Dynamic Limit to the Riemann Solutions of Euler Equations: I. Superposition of rarefaction waves and contact discontinuity

TL;DR

This paper proves the fluid dynamic limit from Navier-Stokes and Boltzmann equations to Euler solutions with complex wave structures, specifically a superposition of two rarefaction waves and a contact discontinuity, including convergence rates.

Contribution

It provides the first rigorous proof of the fluid dynamic limit for a Riemann solution with three superimposed waves, extending previous results for single waves.

Findings

Established convergence of Navier-Stokes and Boltzmann solutions to Euler Riemann solutions with three waves.

Derived uniform convergence rates in terms of viscosity, heat conductivity, and Knudsen number.

First proof of this limit for complex wave superpositions in fluid dynamics.

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.