Abstract

The Landau equation was proposed as the ‘grazing collision limit’ of the Boltzmann equation to account for the singularities in this regime of colliding particles. Inspired by Erbar’s gradient flow description of the Boltzmann equation for Maxwellian kernels, I will present an analogous gradient flow perspective of the Landau equation for Maxwellian and soft potential kernels down to, but excluding the physically relevant, Coulomb kernel. In particular, I will show how the Landau equation can be viewed as the gradient descent of the Boltzmann entropy with respect to a tailor-made metric generalising the dynamic viewpoints of optimal transport metrics from Benamou and Brenier. I hope that this alternative viewpoint of the Landau equation can motivate new ideas and techniques coming from gradient flow theory to attack its open problems. This is joint work with José A. Carrillo, Matias Delgadino, and Laurent Desvillettes.