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arXiv · 2107.08531

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

Katharina Hopf
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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.

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Katharina Hopf. 2021-07-18. Singularities in $L^1$-supercritical Fokker-Planck equations: A qualitative analysis. https://doi.org/10.4171/aihpc%2F85

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