The Annals of Applied Probability

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

Nicolas Fournier

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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 $\mathbb{R}^{3}$.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 860-897.

Dates
First available in Project Euclid: 19 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1424355132

Digital Object Identifier
doi:10.1214/14-AAP1012

Mathematical Reviews number (MathSciNet)
MR3313757

Zentralblatt MATH identifier
1322.82013

Subjects
Primary: 82C40: Kinetic theory of gases 60J75: Jump processes 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Kinetic equations regularization absolute continuity entropy Besov spaces

Citation

Fournier, Nicolas. Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab. 25 (2015), no. 2, 860--897. doi:10.1214/14-AAP1012. https://projecteuclid.org/euclid.aoap/1424355132


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