Singularities in $L^1$-supercritical Fokker-Planck equations: A qualitative analysis

TL;DR

This paper analyzes the formation and evolution of singularities in supercritical nonlinear Fokker-Planck equations, demonstrating convergence to equilibrium and addressing blow-up continuation in a model relevant to Bose-Einstein particles.

Contribution

It provides the first rigorous analysis of singularity formation, continuation beyond blow-up, and long-time behavior for a class of supercritical Fokker-Planck equations, including the 3D Kaniadakis--Quarati model.

Findings

Solutions develop singularities above critical mass.

Solutions converge to the free energy minimizer over time.

The study extends understanding of blow-up and continuation in supercritical regimes.

Abstract

A class of nonlinear Fokker-Planck equations with superlinear drift is investigated in the $L^1$-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis with main focus on the isotropic case, where solutions will be shown to converge to the unique minimiser of the free energy as time tends to infinity. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.