Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric

We study a model Boltzmann equation closely related to the BGK equation using a steepest-descent method in the Wasserstein metric, and prove global existence of energy-and momentum-conserving solutions. We also show that the solutions converge to the manifold of local Maxwellians in the large-time limit, and obtain other information on the behavior of the solutions. We show how the Wasserstein metric is natural for this problem because it is adapted to the study of both the free streaming and the ''collisions''.