The fundamental solution of a nonlinear kinetic Fokker-Planck equation

TL;DR

This paper analyzes the fundamental solution of a nonlinear kinetic Fokker-Planck equation with self-similar behavior, revealing structural properties and asymptotic stability related to nonlinear diffusion phenomena.

Contribution

It extends classical nonlinear diffusion profiles to a kinetic setting, demonstrating preserved self-similar structures and stability properties at the kinetic level.

Findings

Self-similar behavior of the fundamental solution is established.

The kinetic equation exhibits a stability property of a stationary solution.

Structural preservation at the kinetic level impacts entropy estimates and asymptotics.

Abstract

This paper is devoted to a fundamental solution of a nonlinear kinetic equation involving a porous medium or fast diffusion operator acting on velocities. Such a nonlinearity has interesting scaling properties, which result in a self-similar behaviour of the fundamental solution. Here fundamental solution means a Dirac distribution initial datum which moreover governs the large time asymptotics of a large class of solutions. Using a self-similar change of variables, the equation becomes a nonlinear kinetic Fokker-Planck equation with harmonic confinement and the intermediate asymptotics regime is transformed into a stability property of a special stationary solution, which attracts the solutions for large times. In the homogeneous case (pure nonlinear diffusion), the problem is reduced to a classical nonlinear diffusion equation with Barenblatt-Pattle self-similar profiles. Unexpectedly, this beautiful structure is preserved at kinetic level, with remarkable consequences for relative entropy estimates, detailed intermediate asymptotics and nonlinear diffusion limits in adapted functional spaces.