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

In this paper we consider a particular class of solutions of the linear Boltzmann-Rayleighequation, known in the nonlinear setting as homoenergetic solutions. These solutions describethe dynamics of Boltzmann gases under the effect of different mechanical deformations.Therefore, the long-time behaviour of these solutions cannot be described by Maxwelliandistributions and it strongly depends on the homogeneity of the collision kernel of the equa-tion.Here we focus on the paradigmatic case of simple shear deformations and in the case ofcut-off collision kernels with homogeneity gamma >= 0, in particular covering the case of Maxwellmolecules (i.e.gamma=0) and hard potentials with 0 <=gamma<1. We first prove a well-posednessresult for this class of solutions in the space of non-negative Radon measures and then werigorously prove the existence of a stationary solution under the non-equilibrium conditionwhich is induced by the presence of the shear deformation. In the case of Maxwell moleculeswe prove that there is a different behaviour of the solutions for small and large values of theshear parameter