On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:258

Article

Lu, Xuguang¹  Mouhot, Clement² 

[1] Tsinghua Univ, Dept Math Sci, Beijing 100084, Peoples R China

[2] Univ Cambridge, DPMMS, Ctr Math Sci, Cambridge CB3 0WA, England

关键词: Boltzmann equation;    Spatially homogeneous;    Hard potentials;    Measure solutions;    Eternal solution;    Global-in-time stability;   

DOI  :  10.1016/j.jde.2015.01.039

来源: Elsevier

【 摘 要 】

The paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. We prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy. The proof is based on the regularizing property of the iterated collision operators, exponential moment production estimates, and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data. We also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the Maxwellian equilibrium. The constants in these convergence rates depend only on the collision kernel and conserved quantities (mass, momentum, and energy). We finally use these convergence rates in order to deduce global-in-time strong stability of measure solutions. (c) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free