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

Stanley Snelson, Caleb Solomon

Published 2023-09-27Version 1

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. If we specialize our result to the spatially homogeneous case, we recover the best known continuation criterion in that regime.