The Rayleigh-Boltzmann equation with shear deformations in the hyperbolic-dominated regime

In this paper we consider a particular class of solutions of the Rayleigh-Boltzmann equation, known in the nonlinear setting as homoenergetic solutions, which have the form g( x,v,t ) =f( v-L( t)x,t) where the matrix $L(t)$ describes a shear flow deformation. We began this analysis in [22] where we rigorously proved the existence of a stationary non-equilibrium solution and established the different behaviour of the solutions for small and large values of the shear parameter, for cut-off collision kernels with homogeneity parameter 0≤ γ<1, including Maxwell molecules and hard potentials. In this paper, we concentrate in the case where the deformation term dominates the collision term for large times (hyperbolic-dominated regime). This occurs for collision kernels with $γ< 0$ and in particular we focus on γ∈ (-1,0). In such a hyperbolic-dominated regime, it appears challenging to provide a clear description of the long-term asymptotics of the solutions. Here we present a formal analysis of the long-time asymptotics for the distribution of velocities and provide the explicit form for the asymptotic profile. Additionally, we discuss the different asymptotic behaviour expected in the case of homogeneity $γ< -1$. Furthermore, we provide a probabilistic interpretation describing a stochastic process consisting in a combination of collisions and shear flows. The tagged particle velocity (t)≥ 0 is a Markov process that arises from the combination of free flights in a shear flow along with random jumps caused by collisions.