# On the singularity formation and relaxation to equilibrium in 1D Fokker-Planck model with superlinear drift

## TL;DR

This paper analyzes one-dimensional Fokker-Planck equations with superlinear drift, revealing conditions for singularity formation, condensate emergence, and long-term behavior, using a reformulation in pseudo-inverse variables and viscosity solutions.

## Contribution

It introduces a novel framework for global existence, uniqueness, and regularity of solutions in the 1D case, including detailed analysis of blow-up and condensate formation.

## Key findings

- Finite critical mass leads to singular measures
- Solutions blow up in finite time in the supercritical case
- Condensates can be transient and interact with the density

## Abstract

We consider a class of Fokker--Planck equations with linear diffusion and superlinear drift enjoying a formal Wasserstein-like gradient flow structure with convex mobility function. In the drift-dominant regime, the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component. Our approach, which addresses the one-dimensional case, is based on a reformulation of the problem in terms of the pseudo-inverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for global-in-time existence, uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate (resp. singular) parabolic equations, using as a key tool comparison principles and maximum arguments. We then focus on a specific equation and study in more detail the regularity and dynamics of solutions. In particular, blow-up behaviour, formation of condensates (i.e. Dirac measures at zero) and long-time asymptotics are investigated. As a consequence, in the mass-supercritical case, solutions will blow up in $L^\infty$ in finite time and---understood in a generalised, measure sense---they will eventually have condensate. We further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient. The equations considered are motivated by a model for bosons introduced by Kaniadakis and Quarati (1994), which has a similar entropy structure and a critical mass if $d\ge3$.