$L^p$-norms for the homogeneous non-cutoff Boltzmann equation with soft potentials

TL;DR

This paper proves that solutions to the non-cutoff homogeneous Boltzmann equation with soft potentials maintain and develop $L^p$-norms over time, highlighting the role of collision kernel singularity in regularization.

Contribution

It extends previous results on $L^p$-norm propagation from hard to soft potentials in the non-cutoff Boltzmann equation.

Findings

Propagation of $L^p$-norms established

Generation of $L^p$-norms demonstrated

Regularization effects due to collision kernel singularity

Abstract

We establish a priori estimates showing the propagation and generation of $L^p$-norms for solutions to the non-cutoff spatially homogeneous Boltzmann equation with soft potentials. The singularity of the collision kernel is key to generate regularization and inhomogeneity in the energy estimates of the $L^p$-norms. Our result extends \cite{Alo19} from the hard potential cases to the soft ones.