The Limit of the Boltzmann Equation to the Euler Equations for Riemann Problems

TL;DR

This paper investigates the convergence of the Boltzmann equation to the Euler equations in Riemann problems, addressing a longstanding open problem by introducing hyperbolic waves to justify the limit.

Contribution

It provides a rigorous justification of the Boltzmann to Euler limit in complex Riemann solutions using hyperbolic waves to handle different wave interactions.

Findings

Successfully justified the limit for Riemann solutions with shocks, rarefactions, and contact discontinuities.

Introduced hyperbolic waves to capture extra mass transfer in the limit process.

Extended the understanding of kinetic to fluid dynamic limits in complex wave scenarios.

Abstract

The convergence of the Boltzmann equaiton to the compressible Euler equations when the Knudsen number tends to zero has been a long standing open problem in the kinetic theory. In the setting of Riemann solution that contains the generic superposition of shock, rarefaction wave and contact discontinuity to the Euler equations, we succeed in justifying this limit by introducing hyperbolic waves with different solution backgrounds to capture the extra masses carried by the hyperbolic approximation of the rarefaction wave and the diffusion approximation of contact discontinuity.