  In this paper we study a class of solutions of the Boltzmann equation which
have the form $f\left( x,v,t\right) =g\left( v-L\left( t\right) x,t\right) $
where $L\left( t\right) =A\left( I+tA\right) ^{-1}$ with the matrix $A$
describing a shear flow or a dilatation or a combination of both. These
solutions are known as homoenergetic solutions. We prove existence of
homoenergetic solutions for a large class of initial data. For different
choices for the matrix $A$ and for different homogeneities of the collision
kernel, we characterize the long time asymptotics of the velocity distribution
for the corresponding homoenergetic solutions. For a large class of choices of
$A$ we then prove rigorously the existence of self-similar solutions of the
Boltzmann equation. The latter are non Maxwellian distributions and describe
far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for
the $H$-function for some of these flows. These formulas show that in some
cases, despite being very far from equilibrium, the relationship between
density, temperature and entropy is exactly the same as in the equilibrium
case. We make conjectures about the asymptotics of homoenergetic solutions that
do not have self-similar profiles.
