Regularization estimates of the Landau–Coulomb diffusion

The Landau–Coulomb equation is an important model in plasma physics featuring both nonlinear diffusion and reaction terms. In this manuscript we focus on the diffusion operator within the equation by dropping the potentially nefarious reaction term altogether. We show that the diffusion operator in the Landau–Coulomb equation provides a much stronger $L^1\to L^{\infty }$ rate of regularization than its linear counterpart, the Laplace operator. The result is made possible by a nonlinear functional inequality of Gressman, Krieger, and Strain together with a De Giorgi iteration. This stronger regularization rate illustrates the importance of the nonlinear nature of the diffusion in the analysis of the Landau equation and raises the question of determining whether this rate also happens for the Landau–Coulomb equation itself.