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

The grazing collision limit of Kac caricature of Bose-Einstein particles

We discuss the grazing collision limit of certain kinetic models of Bose-Einstein particles obtained from a suitable modification of the one- dimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a nonlinear Fokker-Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in [13 ]. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation.Comment: 28 page

Measure-valued weak solutions for some kinetic equations with singular kernels for quantum particles.

152 p.In this thesis, we present a mathematical study of three problems arising in the kinetic theory of quantumgases.In the first part, we consider a Boltzmann type equation that is used to describe the evolution of theparticle density of a homogeneous and isotropic photon gas, that interacts through Compton scatteringwith a low-density electron gas at non-relativistic equilibrium.Due to the highly singular redistribution function, we consider an approximation that is, nevertheless, stillsingular at the origin. The global existence of measure-valued weak solutions for a large set of initial datais established.We also study a simplified version of this equation, that appears at very low temperatures of the electrongas, where only the quadratic terms are kept. The global existence of measure-valued weak solutions isproved for a large class of initial data, as well as the global existence of $L^1$ solutions for initial datathat satisfy a strong integrability condition. The long time asymptotic behavior of weak solutions for thissimplified equation is also described.In the second part of the thesis, we consider a system of two coupled kinetic equations related to ansimplified model for the evolution of the particle density of the normal and superfluid components in ahomogeneous and isotropic weakly interacting dilute Bose gas.We prove the global existence of measure-valued weak solutions for a large class of initial data. Theconservation of mass and energy and the production of moments of all positive order is also established.Finally, we study some of the properties of the condensate density and we establish an integral equationthat describes its time evolution

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