Decay and Continuity of Boltzmann Equation in Bounded Domains

2008/01/31 by Yan Guo · 2 citations
Mathematics · #math.AP #msc:35Q99

paper · pdf · doi:10.1007/s00205-009-0285-y

89 pages,

arxiv created 2008/04/06 · arxiv updated 2015/05/12

Abstract

Boundaries occur naturally in kinetic equations and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: inflow, bounce-back reflection, specular reflection, and diffuse reflection. We establish exponential decay in L^∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set of the velocity at the boundary. Our contribution is based on a new L² decay theory and its interplay with delicate % L^∞ decay analysis for the linearized Boltzmann equation, in the presence of many repeated interactions with the boundary.

Cited by

• The initial boundary value problem for the Boltzmann equation with soft potential
• Long time behavior of Fokker-Planck equations for bosons and fermions

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