Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation: Collision-Dominated Case

In this paper, we present a formal analysis of the long-time asymptotics of a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, which have the form $f(x,v,t) = g(v-L(t)x,t)$ where $L(t) = A(I+ tA)^{-1}$ with the matrix A describing a shear flow or a dilatation or a combination of both. We began this study in James et al. (Arch Ration Mech Anal 231(2):787–843, 2019). Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. In James et al. (2019), it has been proved rigorously the existence of self-similar solutions which describe the change of the average energy of the particles of the system in the case in which there is a balance between the hyperbolic and the collision term. In this paper, we focus in homoenergetic solutions for which the collision term is much larger than the hyperbolic term (collision-dominated behavior). In this case, the long-time asymptotics for the distribution of velocities is given by a time-dependent Maxwellian distribution with changing temperature.