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The Landau equation
as a gradient Flow
José A. Carrillo, Matias G. Delgadino, Laurent Desvillettes and Jeremy S.-H. Wu |
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Vol. 17 (2024), No. 4, 1331–1375
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Abstract |
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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
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Mathematical Subject Classification
Primary: 35Q82, 49Q22
Secondary: 82C40
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Milestones
Received: 23 October 2021
Revised: 1 June 2022
Accepted: 27 October 2022
Published: 17 May 2024
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Authors |
| José A. Carrillo | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
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| Matias G. Delgadino | |
| Department of Mathematics University of Texas Austin, TX United States |
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| Laurent Desvillettes | |
| Université Paris Cité and Sorbonne
Université, CNRS, IUF, IMJ-PRG Paris, France |
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| Jeremy S.-H. Wu | |
| Department of Mathematics University of California Los Angeles, CA United States |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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