A blow-down mechanism for the Landau-Coulomb equation

TL;DR

This paper demonstrates a blow-down mechanism for the Landau-Coulomb equation, showing solutions decay exponentially to equilibrium and remain close to explicit functions through a novel two-scale linearization approach.

Contribution

It introduces an explicit blow-down function and a new two-scale linearization method for analyzing the Landau-Coulomb equation's solutions.

Findings

Solutions exhibit exponential decay towards equilibrium.

The initial bump region disappears in finite time.

The method provides uniform estimates in the perturbation parameter.

Abstract

We investigate the Landau-Coulomb equation and show an explicit blow-down mechanism for a family of initial data that are small-scale, supercritical perturbations of a Maxwellian function. We establish global well-posedness and show that the initial bump region will disappear in a time of order one. We prove that the function remains close to an explicit function during the blow-down. As a consequence, our result shows exponential decay in time of the solution towards equilibrium. The key ingredients of our proof are the explicit blow-down function and a novel two-scale linearization in appropriate time-dependent spaces that yields uniform estimates in the perturbation parameter.