Local well-posedness for the Boltzmann equation with very soft potential and polynomially decaying initial data

TL;DR

This paper proves local well-posedness for the inhomogeneous non-cutoff Boltzmann equation with very soft potentials and polynomial decay initial data, extending previous results and simplifying proofs using Carleman decomposition.

Contribution

It extends local well-posedness results to very soft potentials without the previous restrictions, using a novel Carleman decomposition approach.

Findings

Complete local well-posedness for $ ext{γ} + 2s < 0$

Simplified proof for $ ext{γ} ext{in} (-3,0]$, $s ext{in} (0,1/2)$

Removes previous decay restriction $ ext{γ} + 2s > -3/2$

Abstract

In this paper, we address the local well-posedness of the spatially inhomogeneous non-cutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials $\gamma + 2s < 0$. Our main result completes the picture for local well-posedness in this decay class by removing the restriction $\gamma + 2s > -3/2$ of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when $\gamma \in (-3,0]$ and $s \in (0,1/2)$ in a weighted $C^1$ space that we include as well.