WIAS Preprint No. 2860, (2021)

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


  • Hopf, Katharina
    ORCID: 0000-0002-6527-2256

2020 Mathematics Subject Classification

  • 35Q84 35K55 35A21 35R06 35B40


  • Fokker--Planck equations for bosons, nonlinear mobility, continuation beyond singularities, singular limit, universal blow-up profile, relaxation to minimising measure




A class of nonlinear Fokker--Planck equations with superlinear drift is investigated in the L1-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 -- the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. 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.

Appeared in

  • Ann. Inst. H. Poincare Anal. Non Lineaire, 41 (2024), pp. 357--403, DOI 10.4171/AIHPC/85 .

Download Documents