Global well-posedness of the Boltzmann equation with large amplitude initial data

TL;DR

This paper proves the global existence and uniqueness of solutions to the Boltzmann equation with large amplitude initial data, using a new approach that handles both hard and soft potentials and analyzes long-term behavior.

Contribution

It introduces a novel $L^ty_xL^1_v \u2229 L^ty_{x,v}$ method to establish global well-posedness for large amplitude initial data in the Boltzmann equation.

Findings

Proves global existence and uniqueness of solutions.

Handles both hard and soft potentials with angular cut-off.

Provides explicit convergence rates for long-time behavior.

Abstract

The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new $L^\infty_xL^1_{v}\cap L^\infty_{x,v}$ approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted $L^\infty$ norm under some smallness condition on $L^1_xL^\infty_v$ norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in $L^\infty_{x,v}$ norm with explicit rates of convergence is also studied.