Regularization estimates and hydrodynamical limit for the Landau equation

TL;DR

This paper establishes new short-time regularization estimates for the nonlinear Landau equation under Navier-Stokes scaling, demonstrating uniform regularity gains and strong convergence to fluid dynamics in a perturbative framework.

Contribution

It introduces optimal, time-quantified regularization estimates for the Landau equation that are uniform in the Knudsen number, linking kinetic and fluid models.

Findings

Obtained instantaneous anisotropic regularity gain for the Landau equation.

Proved uniform regularization estimates in velocity variable.

Established strong convergence towards Navier-Stokes-Fourier system.

Abstract

In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and optimal, indeed, we obtain the instantaneous expected anisotropic gain of regularity (see [53] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system.