Asymptotic behavior of large-amplitude solutions to the Boltzmann equation with soft interactions in $L^p_v L^\infty_x$ spaces

TL;DR

This paper establishes the global existence and convergence to equilibrium of solutions to the Boltzmann equation with soft potentials in an $L^p_v L^ty_x$ setting, overcoming spectral gap issues using a time-involved weight function.

Contribution

It introduces a novel analytical framework with a time-involved weight function and a modified solution operator to handle large-amplitude solutions without spectral gap.

Findings

Proves global well-posedness for large initial data.

Derives sub-exponential decay rates to equilibrium.

Develops new pointwise estimates for nonlinear terms.

Abstract

In this paper, we study the global well-posedness of the Boltzmann equation within the $L_{v}^{p}L_{x}^{\infty}$ framework for soft potential models with angular cutoff in a periodic box $\mathbb{T}^3$. By using a time-involved weight function, inspired by the works of [Liu-Yang,2017], [Duan-Yang-Zhao,2013], [Ko-Lee-Park,2022], we overcome the absence of a spectral gap. An analytical difficulty in the $L_v^p L_x^\infty$ setting is that the standard arguments used in [Ko-Lee-Park,2022], [Li,2022] for the nonlinear loss term are no longer applicable when dealing with time integration involving the collision frequency. To resolve this, we introduce a modified solution operator. Furthermore, we control the nonlinear gain term by deriving pointwise estimates bounded by $L_v^p$ and $L_v^\ell$ (for some $\ell <p$) norms. Thanks to the smallness of the initial relative entropy and Gr\"{o}nwall's inequality, we prove the global existence of unique solutions for large-amplitude initial data and obtain a sub-exponential convergence rate toward equilibrium.