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arXiv · 2501.04399

Time-asymptotic stability of generic Riemann solutions for Boltzmann equation

Yi Wang·Qiuyang Yu
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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.

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Yi Wang, Qiuyang Yu. 2025-01-08. Time-asymptotic stability of generic Riemann solutions for Boltzmann equation. https://arxiv.org/abs/2501.04399

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