On $A_p$ weights and the Landau equation

TL;DR

This paper studies the regularization effects of the Landau equation for different potentials, showing instant smoothness for moderately soft potentials and conditional regularity for very soft potentials, with analysis involving $A_p$-weights.

Contribution

It introduces a novel analysis of the linearized Landau operator using $A_p$-weights, providing new regularity results for Coulomb and other potentials.

Findings

Weak solutions become smooth instantly for moderately soft potentials.

Conditional regularity depends on a nonlinear Morrey space bound.

The linearized operator is a degenerate elliptic Schrödinger operator controlled by $A_p$-weights.

Abstract

In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of moderately soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for moderately soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schr\"odinger operator whose coefficients are controlled by $A_p$-weights.