Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition

Abstract

We consider the $3D$ spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We assume that the initial condition is a probability measure with finite energy and is not a Dirac mass. For hard potentials, we prove that any reasonable weak solution immediately belongs to some Besov space. For moderately soft potentials, we assume additionally that the initial condition has a moment of sufficiently high order ($8$ is enough) and prove the existence of a solution that immediately belongs to some Besov space. The considered solutions thus instantaneously become functions with a finite entropy. We also prove that in any case, any weak solution is immediately supported by ${\mathbb{R}}^3$.

Published in at http://dx.doi.org/10.1214/14-AAP1012 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates
• Coupling property and gradient estimates of Lévy processes via the symbol
• On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

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