Global weak solutions to nonlinear kinetic Fokker--Planck equations in bounded domains under physical initial data

TL;DR

This work addresses global weak solvability for nonlinear kinetic Fokker–Planck equations with degenerate diffusion in bounded domains under physically meaningful data. The authors develop a novel compactness principle based on weighted Fisher information, enabling strong $L^1$ convergence of approximate solutions despite lack of uniform ellipticity. They prove global weak existence for both inflow and partial absorption–reflection boundary conditions, under minimal assumptions on the initial data and boundary data, and establish energy and entropy inequalities along with strong convergence of velocity averages. The approach combines regularization, a fixed-point argument, renormalization, velocity averaging, and a careful limit passage, providing a robust framework that extends weak solvability beyond perturbative regimes and accommodates general diffusion profiles. The results have potential implications for kinetic models with nonlinear collisional terms in bounded geometries and for understanding how physical data alone control global behavior.

Abstract

We establish the global existence of weak solutions to a nonlinear kinetic Fokker--Planck equation with degenerate diffusion, under either inflow or partial absorption-reflection boundary conditions. The novelty of our approach lies in constructing solutions under solely the physical assumptions on the initial and boundary data, namely finite mass, kinetic energy, and entropy, with no additional regularity imposed. To overcome the lack of uniform ellipticity, we develop a new compactness principle based on weighted Fisher information, which yields strong $L^1$ convergence of approximate solutions. This framework provides a robust existence theory under only the physically relevant conditions, and applies uniformly to both inflow and reflection boundary settings.