On the singularity formation and long-time asymptotics in a class of nonlinear Fokker–Planck equations

Abstract

This thesis investigates the properties and long-time behaviour of solutions to a class of Fokker–Planck-type equations with superlinear drift formally dominating the viscous term at high values of the density and potentially leading to the formation of singularities in finite time. The first and main part of this thesis is devoted to a family of Fokker–Planck equations with superlinear drift related to condensation phenomena in quantum physics. 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 the special case of the bosonic Fokker–Planck model in 1D and study in more detail the regularity and dynamics of solutions. In particular, blow-up behaviour, formation of condensates and long-time asymptotics are investigated. We complement the rigorous analysis with numerical experiments enabling conjectures about the condensation process and long-time dynamics in the isotropic 3D Kaniadakis–Quarati model for bosons, the Fokker–Planck equation originally proposed in the physics literature. The simulations suggest that, in the L 1 -supercritical regime, the bosonic Fokker–Planck problem in 1D serves as a good toy model for the Kaniadakis–Quarati model in 3D. The second part of this thesis investigates a question related to fluid mixing and biological cell aggregation. We consider an aggregation equation with fractional (anomalous) diffusion, a generalisation of the classical parabolic-elliptic Keller–Segel system for chemotaxis, which is known to admit solutions exploding in finite time, and study the effect of an ambient incompressible flow on the system. We identify a class of stationary flows significantly enhancing dissipation in the diffusive problem and show that, provided sufficiently strong, these flows are capable of preventing the formation of singularities in our aggregation-diffusion equation and lead to a relaxation to equilibrium at an exponential rate