The Landau equation and Fisher information

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.