Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials

In this paper, we consider a particular class of solutions to the Boltzmann equation which are referred to as homoenergetic solutions. They describe the dynamics of a dilute gas due to collisions and the action of either a shear, a dilation or a combination of both. We prove that solutions with initially high temperature remain close and converge to a Maxwellian distribution with temperature going to infinity. Furthermore, we give precise asymptotic formulas for the temperature. This local stability result is a consequence of a dominant shear and the homogeneity $\gamma>0$ of the collision operator with respect to relative velocities. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both non-cutoff and cutoff kernels.