The Landau equation and Fisher information

TL;DR

This paper demonstrates that Fisher information decreases over time for solutions to the Landau equation with Coulomb potential, using a novel lifted equation approach to rule out finite-time blow-up.

Contribution

It introduces a new lifted linear equation framework that captures the nonlinear collision operator, establishing Fisher information monotonicity for the Landau equation.

Findings

Fisher information is non-increasing along solutions

Lifted equation encodes the collision operator as heat equations

Blow-up is ruled out for Coulomb potential case

Abstract

In this expository note (submitted to Notices of the AMS) we present the ideas used in our recent work ruling out blow up for the Landau equation with Coulomb potential. Blow up is ruled out by the discovery that the Fisher information is not increasing in time along a solution. This monotonicity is established by means of a new ``lifted equation'' which is an auxiliary linear equation in double the number of variables that encodes the nonlinear nonlocal collision operator. For the Landau equation in particular this lifted equation amounts to a family of heat equations over the sphere. Some background on kinetic equations and the Fisher information, and connections to Bakry-Emery theory is also discussed.