The Boltzmann equation in the homogeneous critical regularity framework

TL;DR

This paper establishes the existence, uniqueness, and optimal decay rates of solutions to the 3D Boltzmann equation near equilibrium within a critical Besov space framework, advancing hypocoercivity theory without relying on traditional $L^2$ estimates.

Contribution

It introduces a novel approach to hypocoercivity for the Boltzmann equation using Besov spaces, avoiding $L^2$ estimates and constructing a Lyapunov functional with frequency-dependent dissipation.

Findings

Global unique solutions in critical Besov spaces.

Optimal decay rates for solutions and microscopic parts.

New hypocoercivity framework without $L^2$ estimates.

Abstract

We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space $\widetilde{L}^2_{\xi}(\dot{B}_{2,1}^{1/2}\cap\dot{B}_{2,1}^{3/2})$. In addition, under the condition that the low-frequency part of initial perturbation is bounded in $\widetilde{L}^2_{\xi}(\dot{B}_{2,\infty}^{\sigma_{0}})$ with $-3/2\leq\sigma_{0}<1/2$, it is shown that the solution converges to its equilibrium in large times with the optimal rate of $\mathcal{O}(t^{-(\sigma-\sigma_{0})/2})$ in $\widetilde{L}^2_{\xi}(\dot{B}_{2,1}^{\sigma})$ with some $\sigma>\sigma_0$, and the microscopic part decays at an enhanced rate of $\mathcal{O}(t^{-(\sigma-\sigma_{0})/2-1/2})$. In contrast to [19], the usual $L^2$ estimates are not necessary in our approach, which provides a new understanding of hypocoercivity theory for the Boltzmann equation allowing to construct the Lyapunov functional with different dissipation rates at low and high frequencies. Furthermore, a time-weighted Lyapunov energy argument can be developed to deduce the optimal time-decay estimates.