Decay and Continuity of the Boltzmann Equation in Bounded Domains

• Published in: Archive for rational mechanics and analysis Vol. 197; no. 3; pp. 713 - 809
• Main Author: Guo, Yan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.09.2010; Springer; Springer Nature B.V
• Subjects: Classical Mechanics; Classical statistical mechanics; Complex Systems; Exact sciences and technology; Fluid- and Aerodynamics; Kinetic theory; Mathematical and Computational Physics; Physics; Physics and Astronomy; Statistical physics, thermodynamics, and nonlinear dynamical systems; Theoretical; Convex Domain; Boltzmann Equation; Hard Potential; Linear Boltzmann Equation; Decay Theory; Continuity equations; Boltzmann equation; Grazing incidence; Wave damping; Exponential stability; Confined space; Linearized equation; Edge effect; Decay time; Dilute solution; Modelling; Kinetics; Kinetic theory
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-009-0285-y

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: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the 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 at the boundary. Our contribution is based on a new L 2 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.