One-Dimensional Dissipative Boltzmann Equation: Measure Solutions, Cooling Rate, and Self-Similar Profile

This manuscript investigates the following aspects of the one-dimensional dissipative Boltzmann equation associated to a variable hard-spheres kernel: (1) we show the optimal cooling rate of the model by a careful study of the system satisfied by the solution's moments, (2) we give existence and uniqueness of measure solutions, and (3) we prove the existence of a nontrivial self-similar profile, i.e., homogeneous cooling state, after appropriate scaling of the equation. The latter issue is based on compactness tools in the set of Borel measures. More specifically, we apply a dynamical fixed point theorem on a suitable stable set, for the model dynamics, of Borel measures.