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 (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.

Boltzmann to Landau from the Gradient Flow Perspective

“…More recently, Erbar [23] provided a gradient flow perspective of the Boltzmann equation in the Maxwellian case γ = 0. The current authors and Desvillettes [15] applied a similar strategy for the Landau equation in the soft potential case not including Coulombic interaction γ > −3. Entropy dissipation gradient flow structure.-.…”

“…This notion of solution is known as H-solutions, and one of its salient features is that it only assumes boundedness of relevant physical quantities of the initial data. This perspective was taken further first by Erbar [23] for Boltzmann and then the current authors and Desvillettes [15] for Landau by considering (1.2) as a steepest descent formulation of entropy with a specific 'metric' associated to the dissipation. In these works, the metrics are constructed to rewrite (1.2) as a so-called 'Energy Dissipation (In)equality' (EDI or EDE)…”

“…The quantities | ḟ ǫ | 2 ǫ and | ḟ | 2 L are the metric derivatives with respect to the Boltzmann metric [23] d ǫ , and Landau metric [15] d L . The main contribution of [23] and [15] is to show that, under assumptions on the collision kernel, an absolutely continuous curve f : [0, ∞) → L 1 (R 3 ) with bounded dissipation is a weak solution to Boltzmann or Landau if and only if it satisfies the respective EDI. For this paper, we consider curves f ǫ (t) that satisfy the first EDI of (1.3) as 'H-gradient flow' solutions to the Boltzmann equation and similarly for Landau.…”

“…We recall very formally the grazing collision limit in Section 3.3 which features many similar computations to be repeated later in the paper. The remaining notations and definitions pertaining to abstract gradient flow theory are recalled in Section 4 with an emphasis on the abstract theory developed in [23,15]. We start combining the gradient flow theory with the grazing collision limit in Section 5 which elaborates the compactness mechanisms we use to produce candidate limits for the Landau equation.…”

“…The Landau equation can be derived as the grazing collision limit of the Boltzmann equation, that is when collisions with small angular deviation become predominant. We seek to reformulate the well-known grazing collision relation between the non-cutoff Boltzmann and the Landau equations [18,20,28] from the recently developed gradient flow perspectives [23,15], respectively. For a given collision kernel B, the spatially homogeneous Boltzmann equation in R 3 reads (1.1)…”