Non-existence of some approximately self-similar singularities for the Landau, Vlasov-Poisson-Landau, and Boltzmann equations

TL;DR

This paper proves that certain self-similar singularities do not occur in the Landau, Vlasov-Poisson-Landau, and Boltzmann equations under mild decay conditions, advancing understanding of their solution behaviors.

Contribution

It establishes the non-existence of approximate Type I self-similar blow-up solutions for these kinetic equations, extending previous results to more general potentials and systems.

Findings

No approximate Type I self-similar blow-up solutions exist.

Results apply to very soft and Coulomb potentials.

Analysis includes Vlasov-Poisson-Landau and Boltzmann equations without angular cut-off.

Abstract

We consider the homogeneous and inhomogeneous Landau equation for very soft and Coulomb potentials and show that approximate Type I self-similar blow-up solutions do not exist under mild decay assumptions on the profile. We extend our analysis to the Vlasov-Poisson-Landau system and to the Boltzmann equation without angular cut-off.