Large amplitude solutions in $L^p_vL^\infty_TL^\infty_x$ to the Boltzmann equation for soft potentials

TL;DR

This paper proves the existence of global solutions to the Boltzmann equation with soft potentials in a specific function space, allowing large initial velocity distributions under certain smallness conditions on other norms.

Contribution

It establishes the first global existence result for solutions with large velocity norms in the space $L^p_vL^{ ext{infty}}_T L^{ ext{infty}}_x$ for the soft potential Boltzmann equation.

Findings

Global unique mild solutions exist under specified conditions.

Solutions can have arbitrarily large weighted $L^p_vL^{ ext{infty}}_x$ norms initially.

The proof combines local existence with uniform a priori estimates.

Abstract

In this paper we consider the Cauchy problem on the angular cutoff Boltzmann equation near global Maxwillians for soft potentials either in the whole space or in the torus. We establish the existence of global unique mild solutions in the space $L^p_vL^{\infty}_{T}L^{\infty}_{x}$ with polynomial velocity weights for suitably large $p\leq \infty$, whenever for the initial perturbation the weighted $L^p_vL^{\infty}_x$ norm can be arbitrarily large but the $L^1_xL^\infty_v$ norm and the defect mass, energy and entropy are sufficiently small. The proof is based on the local in time existence as well as the uniform a priori estimates via an interplay in $L^p_vL^{\infty}_{T}L^{\infty}_{x}$ and $L^{\infty}_{T}L^{\infty}_{x}L^1_v$.