Global existence of weak even solutions for an isotropic Landau equation with Coulomb potential

TL;DR

This paper proves the global existence of weak even solutions for an isotropic Landau equation with Coulomb potential in three dimensions, using semi-discretization, entropy inequalities, and moment estimates.

Contribution

It introduces an isotropic modification of the Landau equation and establishes global existence and regularity results for weak solutions with initial data.

Findings

Global existence of weak solutions is established.

Solutions exhibit improved regularity.

Boundedness depends on mass, entropy, and second moment.

Abstract

In this manuscript we consider an isotropic modification for the Landau equation with Coulomb potential in three space dimensions. Global in time existence of weak solutions for even initial data is shown by employing a time semi-discretization of the equation, an entropy inequality and a uniform estimate for the second moment of the solution to the discretized problem. Moreover, under an additional condition that has to be satisfied uniformly over time, uniform boundedness of the solution is proved, with bounds depending solely on the mass, second moment and entropy of the solution. A byproduct of our analysis is a proof of improved regularity for weak solutions to the Landau equation with Coulomb potential.