Abstract

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. More precisely, we study the case in which the shear is dominant compared with the dilation and the collision operator has homogeneity $\gamma >0$. 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. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both noncutoff and cutoff kernels.

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