Dissipation and Semigroup on $H^k_n$: Non-cutoff Linearized Boltzmann Operator with Soft Potential

TL;DR

This paper demonstrates that the linearized non-cutoff Boltzmann collision operator with soft potential generates a strongly continuous semigroup on weighted Sobolev spaces, using pseudo-differential calculus and Gårding's inequality.

Contribution

It establishes the generation of a strongly continuous semigroup for the linearized Boltzmann operator on weighted Sobolev spaces, extending the understanding of dissipation properties in non-cutoff cases.

Findings

The linearized collision operator generates a strongly continuous semigroup on $H^k_n$.

Weighted Sobolev spaces are fundamental in analyzing the Boltzmann equation without angular cutoff.

Pseudo-differential calculus and Gårding's inequality are key tools in the proof.

Abstract

In this paper, we find that the linearized collision operator $L$ of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on $H^k_n$, with $k,n\in\mathbb{R}$. In the theory of Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. The proof is based on pseudo-differential calculus and in general, for a specific class of Weyl quantization, the $L^2$ dissipation implies $H^k_n$ dissipation. This kind of estimate is also known as the G{\aa}rding's inequality.