The large amplitude solution of the Boltzmann equation with soft potential

TL;DR

This paper establishes the existence and asymptotic behavior of large amplitude solutions to the Boltzmann equation with soft potential, using novel velocity weights and modified estimates to handle singularities.

Contribution

It constructs a unique global solution for the soft potential Boltzmann equation with large initial data and small relative entropy, introducing new techniques to manage velocity weights and kernel singularities.

Findings

Global existence of solutions with large amplitude

Sub-exponential decay to equilibrium

Handling of soft potential singularities

Abstract

In this paper, we deal with (angular cut-off) Boltzmann equation with soft potential ($-3<\gamma<0$). In particular, we construct a unique global solution in $L^\infty_{x,v}$ which converges to global equilibrium asymptotically provided that initial data has a large amplitude but with sufficiently small relative entropy. Because frequency multiplier is not uniformly positive anymore, unlike hard potential case, time-involved velocity weight will be used to derive sub-exponential decay of the solution. Motivated by recent development of $L^2\mbox{-}L^\infty$ approach also, we introduce some modified estimates of quadratic nonlinear terms. Linearized collision kernel will be treated in a subtle manner to control singularity of soft potential kernel.