Global stability and scattering theory for non-cutoff Boltzmann equation with soft potentials in the whole space: weak collision regime}

TL;DR

This paper proves the global stability and scattering behavior of traveling Maxwellian solutions to the non-cutoff Boltzmann equation with soft potentials in the whole space, using Strichartz scaling and analytic smoothness propagation.

Contribution

It establishes Lyapunov stability and scattering results for traveling wave solutions in the weak collision regime, with explicit convergence rates in $L^1_{x,v}$ space.

Findings

Traveling Maxwellian is Lyapunov stable.

Perturbed solutions scatter towards a traveling wave.

Explicit convergence rates are obtained.

Abstract

A Traveling Maxwellian $\mathcal{M} = \mathcal{M}(t, x, v)$ represents a traveling wave solution to the Boltzmann equation in the whole space $\R^3_x$(for the spatial variable). The primary objective of this study is to investigate the global-in-time stability of $\mathcal{M}$ and its associated scattering theory in $L^1_{x,v}$ space for the non-cutoff Boltzmann equation with soft potentials when the dissipative effects induced by collisions are {\it weak}. We demonstrate the following results: (i) $\mathcal{M}$ exhibits Lyapunov stability; (ii) The perturbed solution, which is assumed to satisfy the same conservation law as $\mathcal{M}$, scatters in $L^1_{x,v}$ space towards a particular traveling wave (with an explicit convergence rate), which may not necessarily be $\mathcal{M}$. The key elements in the proofs involve the formulation of the {\it Strichartz-Scaled Boltzmann equation}(achieved through the Strichartz-type scaling applied to the original equation) and the propagation of analytic smoothness.