Propagation of moments and sharp convergence rate for inhomogeneous non-cutoff Boltzmann equation with soft potentials

TL;DR

This paper establishes well-posedness and sharp convergence rates for the inhomogeneous non-cutoff Boltzmann equation with soft potentials, demonstrating propagation of moments and convergence to equilibrium in $L^2$ space.

Contribution

It provides the first results on well-posedness and convergence rates for the original equation with soft potentials, using localized techniques and semigroup methods.

Findings

Propagation of exponential moments in $L^2$ space

Sharp convergence rates to the global Maxwellian

Well-posedness for initial data close to Maxwellian

Abstract

We prove the well-posedness for the non-cutoff Boltzmann equation with soft potentials when the initial datum is close to the {\it global Maxwellian} and has only polynomial decay at the large velocities in $L^2$ space. As a result, we get the {\it propagation of the exponential moments} and the {\it sharp rates} of the convergence to the {\it global Maxwellian} which seems the first results for the original equation with soft potentials. The new ingredients of the proof lie in localized techniques, the semigroup method as well as the propagation of the polynomial and exponential moments in $L^2$ space.