On $A_p$ weights and the Landau equation

Maria Gualdani, Nestor Guillen

Published 2017-07-31Version 1

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, the estimates depending merely 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 at an arbitrary and possibly irregular distribution, which is shown to be a Schr\"odinger operator whose coefficients are controlled by $A_p$-weights.