Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential

TL;DR

This paper derives pointwise upper bounds for radially symmetric solutions to the homogeneous Landau equation with Coulomb potential, linking potential blow-up in solutions to concentration phenomena in the $L^{3/2}$-norm.

Contribution

It introduces a novel comparison principle approach to establish bounds and long-time existence results for isotropic Landau equations with Coulomb potential.

Findings

Blow-up in $L^ Infty$-norm linked to $L^{3/2}$-norm concentration.

Provides conditions for long-time existence of solutions.

Establishes bounds using comparison principles for the Landau equation.

Abstract

Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow up in the $L^\infty$-norm at a finite time $T$ can occur only if the $L^{3/2}$-norm of the solution concentrates for times close to $T$. The bounds are obtained using the comparison principle for the Landau equation and for the associated mass function. This method provides long-time existence results for the isotropic version of the Landau equation with Coulomb potential, recently introduced by Krieger and Strain.