Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section

TL;DR

This paper establishes existence, uniqueness, regularity propagation, and stability results for solutions to the Cauchy Boltzmann problem with soft potentials under the Grad cut-off assumption, using an elementary iteration approach.

Contribution

It provides a simplified proof of existence and uniqueness for solutions with large data, including infinite mass, and studies regularity and stability without extra kernel conditions.

Findings

Existence and uniqueness of solutions for large data near Maxwellian regimes.

Propagation of regularity using recent collision operator estimates.

An $L^{p}$-stability result for solutions under the Grad cut-off assumption.

Abstract

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming $S^{n-1}$ integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel--Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes large data near a local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in [3], by E. Carneiro and the authors, that permits to study such propagation without additional conditions on the collision kernel. Finally, an $L^{p}$-stability result (with $1\leq p\leq\infty$) is presented assuming the aforementioned condition.