Time-asymptotic stability of generic Riemann solutions for Boltzmann equation

TL;DR

This paper proves the nonlinear stability of complex wave patterns, including rarefaction, contact discontinuity, and shock, for the one-dimensional Boltzmann equation, addressing a longstanding open problem in kinetic theory.

Contribution

It introduces the first application of the $a$-contraction method to the Boltzmann equation for stability analysis of Riemann solutions.

Findings

Proved nonlinear stability of composite waves for the Boltzmann equation.

Addressed microscopic effects and interactions in shock profiles.

Extended stability results beyond classical fluid dynamics equations.

Abstract

Time-asymptotic stability of generic Riemann solution, consisting of a rarefaction wave, a contact discontinuity and a shock, for the one-dimensional Boltzmann equation, has been a long-standing open problem in kinetic theory. In this paper, we proved that the composite waves of generic Riemann profile including the inviscid self-similar rarefaction wave, the viscous contact wave (i.e., the viscous version of contact discontinuity) and the viscous shock profile with the time-dependent shift to both macroscopic and microscopic components are nonlinearly stable for the one-dimensional Boltzmann equation, by the first using the $a$-contraction method to the Boltzmann equation. Compared with the compressible Navier-Stokes-Fourier equations, the new difficulties here lie in the microscopic effects of the Boltzmann shock profile and their interactions and/or couplings with the rarefaction wave, viscous contact wave and the macroscopic components from the macro-micro decomposition of the Boltzmann equation.