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Time-asymptotic stability of generic Riemann solutions for Boltzmann equation

Yi Wang, Qiuyang Yu

TL;DR

The paper proves the nonlinear time-asymptotic stability of a generic Riemann solution for the one-dimensional Boltzmann equation, consisting of an inviscid self-similar rarefaction wave, a viscous contact wave, and a Boltzmann shock profile with a time-dependent shift. Using an $a$-contraction framework in combination with a macro-micro decomposition, the authors couple macroscopic Navier–Stokes–Fourier-type dynamics with microscopic Boltzmann dissipation; crucially, they incorporate a shift for both macroscopic and microscopic components and handle the intricate interactions between waves and the shock profile. They construct precise wave profiles, derive weighted-energy estimates, and establish global-in-time existence and convergence to the composite wave, with the shift $oldsymbol{X}(t)$ growing sub-linearly and $oldsymbol{\dot X}(t) o0$. This work generalizes stability results from Navier–Stokes–Fourier settings to the Boltzmann equation, highlighting the essential role of microscopic effects in shock stability and providing a rigorous framework for the hydrodynamic limit in kinetic theory. The results have significant implications for kinetic-fluid coupling and the understanding of long-time behavior of near-equilibrium gases under composite wave perturbations.

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.

Time-asymptotic stability of generic Riemann solutions for Boltzmann equation

TL;DR

The paper proves the nonlinear time-asymptotic stability of a generic Riemann solution for the one-dimensional Boltzmann equation, consisting of an inviscid self-similar rarefaction wave, a viscous contact wave, and a Boltzmann shock profile with a time-dependent shift. Using an -contraction framework in combination with a macro-micro decomposition, the authors couple macroscopic Navier–Stokes–Fourier-type dynamics with microscopic Boltzmann dissipation; crucially, they incorporate a shift for both macroscopic and microscopic components and handle the intricate interactions between waves and the shock profile. They construct precise wave profiles, derive weighted-energy estimates, and establish global-in-time existence and convergence to the composite wave, with the shift growing sub-linearly and . This work generalizes stability results from Navier–Stokes–Fourier settings to the Boltzmann equation, highlighting the essential role of microscopic effects in shock stability and providing a rigorous framework for the hydrodynamic limit in kinetic theory. The results have significant implications for kinetic-fluid coupling and the understanding of long-time behavior of near-equilibrium gases under composite wave perturbations.

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 -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.
Paper Structure (18 sections, 19 theorems, 357 equations)

This paper contains 18 sections, 19 theorems, 357 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Approximate rarefaction wave
  4. Viscous contact wave
  5. Boltzmann shock profile
  6. Main result
  7. Proof of main result
  8. Change of variables
  9. Construction of weight function
  10. Construction of shift
  11. A priori estimates
  12. Local-in-time existence
  13. Global-in-time existence
  14. Time-asymptotic behaviors
  15. Properties for the collision operator
  16. ...and 3 more sections

Key Result

Lemma 2.1

(see Xin-1993) Let $\delta_R$ denotes the rarefaction wave strength as $\delta_R := |v_* - v_-|\sim |u_{1*}-u_{1-}|\sim |\theta_*-\theta_-|$. The smooth approximate 1-rarefaction wave $(v^R, u_1^R, \theta^R)(t,x)$ defined in (2.2) satisfies the following properties.

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 1
  • Remark 2
  • Theorem 2.4
  • Remark 3
  • Remark 4
  • Lemma 3.1
  • Proposition 3.1
  • ...and 15 more