Homoenergetic solutions for the Rayleigh-Boltzmann equation: existence of a stationary non-equilibrium solution

TL;DR

This paper investigates Homoenergetic solutions of the linear Boltzmann-Rayleigh equation under shear deformation, establishing existence of stationary non-equilibrium solutions and analyzing their behavior for Maxwell molecules and other potentials.

Contribution

It proves well-posedness and existence of stationary solutions for Homoenergetic solutions with shear deformation in the Boltzmann-Rayleigh equation, covering Maxwell molecules and hard potentials.

Findings

Existence of stationary non-equilibrium solutions under shear deformation.

Different solution behaviors for small and large shear parameters in Maxwell molecules.

Well-posedness of solutions in the space of non-negative Radon measures.

Abstract

In this paper we consider a particular class of solutions of the linear Boltzmann-Rayleigh equation, known in the nonlinear setting as Homoenergetic solutions. These solutions describe the dynamics of Boltzmann gases under the effect of different mechanical deformations. Therefore, the long-time behaviour of these solutions cannot be described by Maxwellian distributions and it strongly depends on the homogeneity of the collision kernel of the equation. Here we focus on the paradigmatic case of simple shear deformations and in the case of cut-off collision kernels with homogeneity $\gamma\geq 0$, in particular covering the case of Maxwell molecules (i.e. $\gamma=0$) and hard potentials with $0\leq \gamma <1$. We first prove a well-posedness result for this class of solutions in the space of non-negative Radon measures and then we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the shear deformation. In the case of Maxwell molecules we prove that there is a different behaviour of the solutions for small and large values of the shear parameter.