On smooth solutions to the thermostated Boltzmann equation with deformation

TL;DR

This paper studies smooth, non-Maxwellian steady solutions to a thermostated Boltzmann equation with deformation, establishing their existence, properties, and long-term behavior under small deformation forces.

Contribution

It constructs smooth steady solutions for the thermostated Boltzmann equation with deformation and analyzes their asymptotic stability and tail behavior.

Findings

Existence of smooth steady solutions for small deformation

Steady solutions are non-Maxwellian with polynomial tails

Long-time asymptotics show convergence to steady states

Abstract

This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions.