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In this paper we continue the formal analysis of the long-time asymptotics ofthe homoenergetic solutions for the Boltzmann equation that we began in [18].They have the form $f\left( x,v,t\right) =g\left(v-L\left( t\right) x,t\right)$ where $L\left( t\right) =A\left(I+tA\right) ^{-1}$ where $A$ is a constantmatrix. Homoenergetic solutions satisfy an integro-differential equation whichcontains, in addition to the classical Boltzmann collision operator, a linearhyperbolic term. Depending on the properties of the collision kernel thecollision and the hyperbolic terms might be of the same order of magnitude as$t\to\infty$, or the collision term could be the dominant one for large times,or the hyperbolic term could be the largest. The first case has been rigorouslystudied in [17]. Formal asymptotic expansions in the second case have beenobtained in [18]. All the solutions obtained in this case can be approximatedby Maxwellian distributions with changing temperature. In this paper we focusin the case where the hyperbolic terms are much larger than the collision termfor large times (hyperbolic-dominated behavior). In the hyperbolic-dominatedcase it does not seem to be possible to describe in a simple way all the longtime asymptotics of the solutions, but we discuss several physical situationsand formulate precise conjectures. We give explicit formulas for therelationship between density, temperature and entropy for these solutions.These formulas differ greatly from the ones at equilibrium.

Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case