The Landau equation as a Gradient Flow

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. 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.

arXiv comment46 pages, accepted for publication

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• Random batch particle methods for the homogeneous Landau equation
• Dynamical large deviations for plasmas below the Debye length and the Landau equation
• Boltzmann to Landau from the Gradient Flow Perspective
• A gradient flow perspective on McKean-Vlasov equations in econophysics
• Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials
• On the gradient flow structure of the isotropic Landau equation