Optimal time decay of the non cut-off Boltzmann equation in the whole space

TL;DR

This paper establishes the optimal large-time decay rates for solutions to the non cut-off Boltzmann equation in the whole space, addressing a longstanding open problem for soft potentials.

Contribution

It proves the decay rates of solutions to the non cut-off Boltzmann equation in the whole space, extending previous results to soft potentials without angular cut-off.

Findings

Solutions converge to Maxwellian at optimal decay rates

Decay rate of $O(t^{-rac{N}{2}+rac{N}{2r}})$ in $L^2_v(L^r_x)$-norm

Addresses longstanding open problem for soft potentials

Abstract

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space $\threed_x$ with $\DgE$. We use the existence theory of global in time nearby Maxwellian solutions from \cite{gsNonCutA,gsNonCut0}. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption \cite{MR677262,MR2847536}. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\frac{\Ndim}{2}+\frac{\Ndim}{2r}})$ in the $L^2_\vel(L^r_x)$-norm for any $2\leq r\leq \infty$.