Space-time behavior of the solution to the Boltzmann equation with soft potentials

TL;DR

This paper analyzes the space-time decay of solutions to the Boltzmann equation with soft potentials near equilibrium, revealing how initial velocity decay and potential exponent influence long-term behavior without requiring Sobolev regularity.

Contribution

It introduces a new approach to directly obtain $L^{ abla} $ bounds for the Boltzmann equation, bypassing traditional Green's function methods.

Findings

Space-time behavior depends on initial velocity decay and potential exponent.

Established $L^{ abla} $ bounds without Sobolev regularity assumptions.

Provides new insights into the decay properties of solutions with soft potentials.

Abstract

In this paper, we get the quantitative space-time behavior of the full Boltzmann equation with soft potentials ($-2<\gamma <0$) in the close to equilibrium setting, under some velocity decay assumption, but without any Sobolev regularity assumption on the initial data. We find that both the large time and spatial behaviors depend on the velocity decay of the initial data and the exponent $\gamma$. The key step in our strategy is to obtain the $L^{\infty }$ bound of a suitable weighted full Boltzmann equation directly, rather than using Green's function and Duhamel's principle to construct the pointwise structure of the solution as in the paper: T.-P. Liu and S.-H. Yu, The Green function and large time behavier of solutions for the one-dimensional Boltzmann equation, Commun. Pure App. Math.,(2004). This provides a new thinking in the related study.