Asymptotic behaviour for equidispersive solutions of the Boltzmann equation

In this talk we consider particular solutions of the Boltzmann equation which have the form $f (x,v,t) = g (v − M (t)x,t)$ where $M (t) = A(I + tA)^{−1}$ with the matrix $A$ describing a shear flow or a dilatation or a combination of both. These solutions are known as equidispersive solutions. We will show that, for different choices for the matrix A and for different homogeneities of the collision kernel, we obtain different long time asymptotics for the corresponding equidispersive solutions. In particular we will focus on the case of simple shear flow and prove rigorously the existence of self-similar solutions with exponentially increasing internal energy.