Sharp regularity and Cauchy problem of the spatially homogeneous Boltzmann equation with Debye-Yukawa potential

TL;DR

This paper investigates the regularity and existence of solutions to the spatially homogeneous Boltzmann equation with Debye-Yukawa potential, demonstrating unique solutions with improved regularity over time.

Contribution

It introduces a spectral decomposition approach to prove existence, uniqueness, and sharp regularity results for solutions with initial data in dual Sobolev spaces.

Findings

Existence of unique solutions for initial data in dual Sobolev spaces.

Solutions gain regularity for any positive time.

Established sharp regularity properties of solutions.

Abstract

In this paper, we study the Cauchy problem for the linear spatially homogeneous Boltzamnn equation with Debye-Yukawa potential. Using the spectral decomposition of the linear operator, we prove that, for an initial datum in the sense of distribution which contains the dual of the Sobolev spaces, there exists a unique solution which belongs to a more regular Sobolev space for any positive time. We also study the sharp regularity of the solution.