Condensation phenomena in nonlinear drift equations

We study nonnegative, measure-valued solutions to nonlinear drift type equations modelling concentration phenomena related to Bose-Einstein particles. In one spatial dimension, we prove existence and uniqueness for measure solutions. Moreover, we prove that all solutions blow up in finite time leading to a concentration of mass only at the origin, and the concentrated mass absorbs increasingly the mass converging to the total mass as time goes to infinity. Our analysis makes a substantial use of independent variable scalings and pseudo-inverse functions techniques

Spectral methods in general relativistic astrophysics

We present spectral methods developed in our group to solve three-dimensional partial differential equations. The emphasis is put on equations arising from astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures, submitted to Journal of Computational & Applied Mathematic

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

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

On a Boltzmann equation for Compton scattering, from non relativistic electrons at low density.

A Boltzmann equation, used to describe the Compton scattering in the non-relativistic limit is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions is proved for a large set of initial data. A simplified equation, where only the quadratic terms are kept, is also studied. The existence of weak solutions, and also of more regular solutions that are very flat near the origin, is proved. The long time asymptotic behavior of weak solutions of the simplified equation is described.MTM2014-52347- C2-1-R of DGE

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

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

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

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