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

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