We consider the three-dimensional Ginzburg-Landau model for a spherical superconductor in a uniform applied field, in the limit as the Ginzburg-Landau parameter tends to infinity. We derive a reduced limiting energy for vortex curves when the applied field is of the order of the logarithm of the Ginzburg-Landau parameter. We show that the global minimizer of this limiting energy must be either the diameter (along the field direction) or the vortexless (Meissner) configuration, depending on the strength of the applied field. For the full energy we show that there exists locally minimizing solutions of the Ginzburg-Landau equations whose vortices converge (in a sense of rectifiable currents) to the diameter when the field is in the range predicted by the analysis of the limiting problem.
This represents joint work with L. Bronsard and J. A. Montero.
After the approach of Duchon and Robert on the Kelvin Helmoltz who considered it as a "Dirichlet" problem, the results of Lebeau and Kamotski on the analyticity of the curve which may carry singularities leads to new ideas for the weak solutions of Kelvin Helmholtz and the Raleigh Taylor problems. Appearance of singularities, breaking of the curve which carries the density of vorticity, necessity of the surface tension, etc. ...
We consider a two-dimensional model for a rotating Bose-Einstein condensate (BEC) in an anharmonic trap. The special shape of the trapping potential, negative in a central hole and positive in an annulus, favors an annular shape for the support of the wave function. We study the minimizers of the energy in the Thomas-Fermi limit for two different regimes of the rotational speed.
For bounded rotations we observe that the energy minimizers acquire vorticity beyond a critical rotational value, but the vortices are strongly pinned in the central hole where the potential is negative. In this regime, minimizers exhibit no vortices in the annular bulk of the condensate. There is a critical rotational speed, which grows as the logarithm of the small parameter, for which this strong pinning effect breaks down and vortices begin to appear in the annular bulk. We derive an asymptotic formula for the critical speed, and determine precisely the location of nucleation of the vortices at the critical value.
This represents joint work with A. Aftalion and S. Alama.
This work addresses the question of deriving hydrodynamic and diffusion models from a macroscopic limit of quantum kinetic models. This question is of key importance in a certain number of fields such as plasma or semiconductor mesoscopic modeling.
The major difficulty to solve when investigating hydrodynamic limits is that of the closure relation (i.e. finding the equation-of-state of the system). This problem is resolved in the classical framework by assuming that the microscopic state is at local thermodynamical equilibrium. Such a state realizes the minimum of the entropy functional subject to local constraints of mass, momentum and energy.
We propose an extension of this method to quantum systems. This leads to hydrodynamic models with non-local closure relations. These models preserve the monotony of the entropy functional. The same approach leads to a proposal for quantum extensions of the classical Boltzmann or BGK collision operators. Finally, it allows the investigation of diffusion limits of quantum systems (which are distinguished from hydrodynamic limits by the nature of the scaling) and leads to quantum extension of the well-established drift-diffusion and energy-transport models.
By studying the evolution of the free-internal, potential and interactive-energies of an interacting system of particles, along the geodesics of mass transport, one can recover many of the basic ingredients of modern analysis (functional inequalities) in a unifying framework that gives a good introduction to several natural evolution equations of Fokker-Planck type. Does it all mean that much of analysis is yet to be discovered?
The Jacobian estimates mentioned in the title of this talk are estimates that control the Jacobian of a (typically complex-valued) function in certain negative Sobolev norms by its Ginzburg-Landau energy. Some model such estimates will be surveyed, and some applications sketched. The remainder of the talk will present new refined Jacobian estimates that are nearly sharp in certain situations of interest in PDE applications.
This talk will briefly review the solution of the inverse boundary value problem of Calderon, and describe analogous questions for quasilinear and semilinear operators.
For a general class of nonlinear, inhomogeneous Schroedinger equations in a bounded planar domain, we show that the nonlinear potential can be analytically reconstructed from knowledge of the corresponding Dirichlet-to-Neumann map on the boundary. This is joint work with Victor Isakov.
The premixed part of a Bunsen burner flame can be modelled-in a very crude approximation-by a reaction-diffusion equation in the plane with conical conditions at infinity. This means that the fresh gases are located in some given cone of the lower half plane. Travelling fronts to such an equation, whose velocity is given by the (100-year-old) Gouy formula, can be shown to exist.
It turns out that the same approach can be carried out successfully in bistable equations, extending an earlier result of P. Fife (concerning almost planar fronts for scalar equations), and more recent results of Haragus-Scheel (almost planar fronts for systems). Our results are valid in the 2D and 3D cylindrically symmetric cases.
Joint work with F. Hamel and R. Monneau.
The Kadomstsev-Petviahvili (KP) equations are universal models to describe the dynamics of long dispersive weakly nonlinear waves propagating in one direction with weak transverse effects. There are two versions, the (focusing) KP I equation, and the (defocusing) KP II equation.
It has been discovered recently (Molinet, Saut, Tzvetkov) that the KP I equation has a "quasilinear" behavior. In particular, contrary to the KP II equation, it cannot be solved by Picard iteration in any natural Sobolev class. This makes the Cauchy problem for KP I quite challenging.
In this talk we will survey recent results on the global Cauchy problem for KP I, due to L. Molinet, N. Tzvetkov and the lecturer, and to C. Kenig. We will in particular solve the Cauchy problem in the background of a line soliton.
According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator D0 (this model is called the "Dirac sea"). In the presence of an external field, these virtual particles react and the vacuum becomes polarized.
In this work, we consider a nonlinear model of the vacuum derived from QED, called the Bogoliubov-Dirac-Fock model (BDF). In this model, the vacuum is represented by a bounded self-adjoint operator G on L2 (R3). An energy of this vacuum is defined. We show the existence of a minimizer of the BDF energy in the presence of an external electrostatic field. Then we prove that this minimizer is a projector, which solves a self-consistent equation of Hartree-Fock type. This minimizer is interpreted as the polarized Dirac sea.
This is joint work with Christian Hainzl and Mathieu Lewin.
The Gross-Pitaevskii equation, a nonlinear Schroedinger equation with non-zero boundary conditions, models superfluids and Bose-Einstein condensates. Recent mathematical work has focused on the short-time dynamics of vortex solutions, and existence of vortex-pair traveling waves. However, little seems to be known about the long-time behaviour (eg. scattering theory, and the asymptotic stability of vortices). We address the simplest such problem-scattering around the vacuum state-which is already tricky due to the non-self-adjointness of the linearized operator, and "long-range" nonlinearity. In particular, our present methods are limited to higher-dimension. This is joint work in progress with S. Gustafson and K. Nakanishi.