Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff

TL;DR

This paper proves that weak solutions to the spatially homogeneous Boltzmann equation without angular cutoff become infinitely smooth in velocity over time, given finite moments of all orders.

Contribution

It demonstrates the smoothing effect for weak solutions of the Boltzmann equation without angular cutoff, a result previously unknown for such solutions.

Findings

Weak solutions become $C^$ smooth in velocity for positive time

Finite moments of all orders lead to regularity gain

Results apply to solutions with initial $L^1$ integrability

Abstract

In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every $L^1$ weak solution to the Cauchy problem with finite moments of all order acquires the $C^\infty$ regularity in the velocity variable for the positive time.