On the Cauchy problem of the Boltzmann equation with a very soft potential

TL;DR

This paper investigates the global existence and decay properties of solutions to the Boltzmann equation with very soft potentials, extending previous methods to cases where the spectral gap is absent and using weighted velocity spaces.

Contribution

It generalizes the estimate on the linearized collision operator to very soft potentials and establishes algebraic decay without spectral gap assumptions.

Findings

Proves global existence for b3a0a0 in [0,d)

Obtains algebraic decay in time for solutions

Extends previous results to very soft potentials without spectral gap

Abstract

The Cauchy problem for the Boltzmann equation with soft potential, in the framework of small perturbation of an equilibrium state, has been studied in many spaces. The method of strongly continuous semigroup has been applied by Caflisch\cite{Caflisch1980a} and Ukai-Asano\cite{Ukai1982} for the case of soft potential, where they obtained the $L^\infty$ solution without requiring any velocity deviation. By generalizing the estimate on linearized collision operator $L$ to the case of very soft potential, we obtain a similar global existence result for $\gamma\in[0,d)$. For soft potential, the spectrum structure of the linearized Boltzmann operator couldn't give spectral gap, so we use the method of integration by parts and consider a weighted velocity space in order to obtain algebraic decay in time.