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On the singularity formation and relaxation to equilibrium in 1D Fokker-Planck model with superlinear drift
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Carrillo, Jose Antonio, Hopf, Katharina and Rodrigo Diez, José L. (2020) On the singularity formation and relaxation to equilibrium in 1D Fokker-Planck model with superlinear drift. Advances in Mathematics, 360 . 106883. doi:10.1016/j.aim.2019.106883 ISSN 0001-8708.
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Official URL: https://doi.org/10.1016/j.aim.2019.106883
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∞ 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≥3.
Item Type: | Journal Article | ||||||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Journal or Publication Title: | Advances in Mathematics | ||||||||||||
Publisher: | Academic Press | ||||||||||||
ISSN: | 0001-8708 | ||||||||||||
Official Date: | 22 January 2020 | ||||||||||||
Dates: |
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Volume: | 360 | ||||||||||||
Article Number: | 106883 | ||||||||||||
DOI: | 10.1016/j.aim.2019.106883 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 5 November 2019 | ||||||||||||
Date of first compliant Open Access: | 12 November 2020 | ||||||||||||
RIOXX Funder/Project Grant: |
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