Communications in Mathematical Sciences

Volume 19 (2021)

Number 8

On the gradient flow structure of the isotropic Landau equation

Pages: 2319 – 2333



Jing An (Institute for Computational and Mathematical Engineering, Stanford University, Stanford, California, U.S.A.)

Lexing Ying (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


We prove that the isotropic Landau equation equipped with the Coulomb potential, introduced by Krieger–Strain and Gualdani–Guillen, can be identified with the gradient flow of the entropy in the probability space with respect to a Riemannian metric tensor with nonlocal mobility. We give characterizations of the corresponding geodesics equations and present a convergence rate result by estimating its Hessian operator.


isotropic Landau equation, nonlocal mobility, gradient flow

2010 Mathematics Subject Classification

35Q20, 35Q70

J.A. is supported by the Joe Oliger Fellowship from Stanford University. The work of L.Y. is partially supported by the National Science Foundation under award DMS-1818449.

Received 4 December 2019

Accepted 3 June 2021

Published 7 October 2021