Non-cutoff Boltzmann equation with soft potentials in the whole space

TL;DR

This paper establishes the existence and uniqueness of global solutions to the non-cutoff Boltzmann equation with soft potentials in the whole space, using a novel decomposition approach for initial data close to Maxwellian.

Contribution

It introduces a new method decomposing solutions into parts with different decay properties, advancing understanding of the non-cutoff Boltzmann equation with soft potentials.

Findings

Proves global existence and uniqueness of solutions.

Develops a decomposition technique for solutions.

Handles polynomial and Gaussian decay in velocity.

Abstract

We prove the existence and uniqueness of global solutions to the Boltzmann equation with non-cutoff soft potentials in the whole space when the initial data is a small perturbation of a Maxwellian with polynomial decay in velocity. Our method is based in the decomposition of the desired solution into two parts: one with polynomial decay in velocity satisfying the Boltzmann equation with only a dissipative part of the linearized operator ; the other with Gaussian decay in velocity verifying the Boltzmann equation with a coupling term.