Local Existence for the Spatially Homogeneous Boltzmann Equation with Soft Potentials

TL;DR

This paper establishes local-in-time existence and uniqueness of smooth solutions for the spatially homogeneous Boltzmann equation with soft potentials, using bilinear estimates for the collision operator.

Contribution

It provides the first rigorous proof of local existence for this class of Boltzmann equations with soft potentials, including bounds on the maximal existence time.

Findings

Proves local-in-time existence and uniqueness of solutions.

Derives lower bounds for the maximal existence time.

Identifies conditions for finite time extinction of solutions.

Abstract

We prove a local-in-time existence and uniqueness theorem for a smooth classical solution to the spatially homogeneous Boltzmann equation with cutoff soft potentials. Our proof is based on a series of bilinear estimates for the integrability and Sobolev regularity of the associated collision operator. While the global-in-time existence is left inconclusive, we give a lower bound of the maximal time of existence and a necessary condition for finite time extinction of existence.