Uniqueness and global dynamics of spatially homogeneous non cutoff Boltzmann equation with moderate soft potentials

TL;DR

This paper establishes the uniqueness, smoothing effects, and long-term behavior of solutions to the spatially homogeneous non-cutoff Boltzmann equation with moderate soft potentials, linking solution regularity and convergence rates to initial velocity decay.

Contribution

It provides the first quantitative characterization of solution regularity and convergence rates based on initial velocity decay for this class of Boltzmann equations.

Findings

Polynomial decay in initial data leads to polynomial convergence to equilibrium.

Exponential decay in initial data results in smooth regularization and stretched exponential convergence.

Development of localized phase and frequency space techniques for analysis.

Abstract

Departing from the weak solution, we prove the uniqueness, smoothing estimates and the global dynamics for the non cutoff spatially homogeneous Boltzmann equation with moderate soft potentials. Our results show that the behavior of the solution(including the production of regularity and the longtime behavior) can be {\it characterized quantitatively} by the initial data at the large velocities, i.e.(i). initially polynomial decay at the large velocities in $L^1$ space will induce the finite smoothing estimates in weighted Sobolev spaces and the polynomial convergence rate (including the lower and upper bounds) to the equilibrium; (ii). initially the exponential decay at the large velocities in $L^1$ space will induce $C^\infty$ regularization effect and the stretched exponential convergence rate. The new ingredients of the proof lie in the development of the localized techniques in phase and frequency spaces and the propagation of the exponential momentum.