A continuation criterion for the Landau equation with very soft and Coulomb potentials

TL;DR

This paper establishes a continuation criterion for solutions to the inhomogeneous Landau equation with very soft and Coulomb potentials, based on the boundedness of mass density, small velocity moments, and certain $L^p$ norms, without needing energy bounds.

Contribution

It provides the first continuation criterion for the Landau equation with Coulomb potentials that does not rely on energy density bounds.

Findings

Solutions can be continued if mass density, small velocity moments, and specific $L^p$ norms remain finite.

No energy density bounds are required for continuation.

The criterion applies uniformly in time and space.

Abstract

We consider the spatially inhomogeneous Landau equation in the case of very soft and Coulomb potentials, $\gamma \in [-3,-2]$. We show that solutions can be continued as long as the following three quantities remain finite, uniformly in $t$ and $x$: (1) the mass density, (2) the velocity moment of order $s$ for any small $s>0$, and (3) the $L^p_v$ norm for any $p>3/(5+\gamma)$. In particular, we do not require a bound on the energy density.