A Gradient Flow Approach to Kinetic Equations

Explore a gradient flow approach to kinetic equations in this 58-minute lecture from the Hausdorff Trimester Program on Kinetic Theory. Delve into the research conducted by Giada Basile, D. Benedetto, and L. Bertini, focusing on a gradient flow formulation of linear kinetic equations using an entropy dissipation inequality. Examine the relationship between this formulation and the large deviation principle for continuous time Markov chains, and consider potential extensions to non-linear cases. Learn about the current-density formulation of the heat equation, linear Boltzmann equations, and their probabilistic interpretations. Investigate the Boltzmann-Grad limit of the Lorentz gas, Rayleigh-Boltzmann equation, and linear phonon Boltzmann equation. Explore the current-measure formulation of LBE, including assumptions, existence, and uniqueness. Dive into topics such as empirical measure and current of N independent copies, continuity equation, large deviation principle, and microscopic models. Conclude with an examination of the rate functional, entropy dissipation inequality, Dirichlet integral, and kinematic term, culminating in a comprehensive gradient flow formulation.

Intro
Contents
A current-density formulation of the heat equation
Variational characterization
Heat equation in probability space
Comments
Linear Boltzmann equations
Probabilistic interpretation
Boltzmann-Grad limit of the Lorentz gas
Rayleigh-Boltzmann equation
Linear phonon Boltzmann equation
A current-measure formulation of LBE
Assumptions
Existence ad uniqueness
Definition of LBE
Remarks
Empirical measure and current of N independent copies
Continuity equation
Large deviation principle
Microscopic model
The collision kernel
Large deviations (work in progress)
The rate functional
Entropy dissipation inequality
The Dirichlet integral and the the kinematic term
A gradient flow formulation