Long-Time Asymptotics of Kinetic Models of Granular Flows

Abstract

We analyze the long-time asymptotics of certain one-dimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi-elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending on whether or not the temperature of their similarity solutions (homogeneous cooling states) reduce to zero in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Wasserstein metric of any two solutions with the same mass and mean velocity. Furthermore, if the temperature of the similarity solution decays to zero in finite time, we prove, by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support, that the temperature of any other solution with initially bounded support must also decay to zero in finite time.