Local existence, lower mass bounds, and a new continuation criterion for the Landau equation

TL;DR

This paper proves short-term existence, mass spreading, and a new continuation criterion for solutions to the inhomogeneous Landau equation with soft potentials, including Coulomb interactions, using stochastic methods and regularity results.

Contribution

It establishes the first local existence results with Gaussian decay assumptions, proves optimal mass lower bounds, and introduces a minimal continuation criterion based on bounded mass and energy densities.

Findings

Solutions instantaneously spread mass to all points.

Mass lower bounds decay sub-Gaussianly, proven to be optimal.

Solutions become smooth in all variables under certain conditions.

Abstract

We consider the spatially inhomogeneous Landau equation with soft potentials, including the case of Coulomb interactions. First, we establish the existence of solutions for a short time, assuming the initial data is in a fourth-order Sobolev space and has Gaussian decay in the velocity variable (no decay assumptions are made in the spatial variable). Next, we show that the evolution instantaneously spreads mass to every point in its domain. The resulting pointwise lower bounds have a sub-Gaussian rate of decay, which we show is optimal. The proof of mass-spreading is based on a stochastic process associated to the equation, and makes essential use of nonlocality. By combining this theorem with prior regularity results, we derive two important applications: $C^\infty$ smoothing in all three variables, even for initial data with vacuum regions, and a continuation criterion that states the solution can be extended for as long as the mass and energy densities stay bounded from above. This is the weakest condition known to prevent blow-up. In particular, it does not require a lower bound on the mass density or an upper bound on the entropy density.