A Continuation Criterion for the Landau Equation with Very Soft and Coulomb Potentials

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.