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Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles

Research paper by Tai-Ping Liu, Shih-Hsien Yu

Indexed on: 29 Jan '04Published on: 29 Jan '04Published in: Communications in Mathematical Physics



Abstract

We introduce an elementary energy method for the Boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components. The decomposition is useful for the study of time-asymptotic stability of nonlinear waves. The wave location is determined by the macroscopic equation. The microscopic component has an equilibrating property. The coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the Chapman-Enskog expansion. Our main result is the establishment of the positivity of shock profiles for the Boltzmann equation. This is shown by the time-asymptotic approach and the maximal principle for the collision operator.