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The Landau equation as a gradient Flow

José A. Carrillo, Matias G. Delgadino, Laurent Desvillettes and Jeremy S.-H. Wu

Vol. 17 (2024), No. 4, 1331–1375
Abstract

We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the energy dissipation inequality (EDI). Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation.

Keywords
Landau equation, gradient flow, steepest descent
Mathematical Subject Classification
Primary: 35Q82, 49Q22
Secondary: 82C40
Milestones
Received: 23 October 2021
Revised: 1 June 2022
Accepted: 27 October 2022
Published: 17 May 2024
Authors
José A. Carrillo
Mathematical Institute
University of Oxford
Oxford
United Kingdom
Matias G. Delgadino
Department of Mathematics
University of Texas
Austin, TX
United States
Laurent Desvillettes
Université Paris Cité and Sorbonne Université, CNRS, IUF, IMJ-PRG
Paris, France
Jeremy S.-H. Wu
Department of Mathematics
University of California
Los Angeles, CA
United States

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