Non-Cutoff Boltzmann Equation with Polynomial Decay Perturbation

TL;DR

This paper establishes well-posedness for the non-cutoff Boltzmann equation with algebraic decay initial data, leveraging spectral analysis, moment propagation, and a novel pseudo-differential operator to handle singularities.

Contribution

It introduces a new well-posedness framework for the non-cutoff Boltzmann equation with polynomial decay, combining spectral gap analysis and a specialized pseudo-differential operator.

Findings

Proved well-posedness for small perturbations of Maxwellian with algebraic decay

Demonstrated regularizing effects in Sobolev spaces with negative indices

Established coercivity estimates using intrinsic symmetries

Abstract

The Boltzmann equation without an angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian with an algebraic decay in the velocity variable. A well-posedness theory in the perturbative framework is obtained for both mild and strong angular singularities by combining three ingredients: the moment propagation, the spectral gap of the linearized operator, and the regularizing effect of the linearized operator when the initial data is in a Sobolev space with a negative index. A carefully designed pseudo-differential operator plays an central role in capturing the regularizing effect. Moreover, some intrinsic symmetry with respect to the collision operator and an intrinsic functional in the coercivity estimate are essentially used in the commutator estimates for the collision operator with velocity weights.