SS6 - Équations aux dérivées partielles / SS6 - Partial Differential Equations
Org: M. Esteban (Paris) et/and C. Sulem (Toronto)
- STAN ALAMA, McMaster University, Hamilton, Ontario, Canada
On the Ginzburg-Landau model of a superconducting sphere in a
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.
- CLAUDE BARDOS, Laboratoire Jacques-Louis Lions, Université Denis Diderot,
175 avenue du Chevaleret, Paris 75013
Applications of regularity results of Lebeau and Kamotsky to
the understanding of the Kelvin Helmoltz and Rayleigh Taylor
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. ...
- LIA BRONSARD, McMaster University, Hamilton, Ontario, Canada
Giant vortex and the breakdown of strong pinning in a
rotating Bose-Einstein condensate
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
This represents joint work with A. Aftalion and S. Alama.
- PIERRE DEGOND, MIP, CNRS and Université Paul Sabatier
Quantum hydrodynamics and diffusion models derived from the
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
- NASSIF GHOUSSOUB, UBC
The optimal evolution of the free energy of interacting gases
and its applications
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?
- ROBERT JERRARD, University of Toronto, Toronto, ON M5S 3G3, Canada
Refined Jacobian estimates for a Ginzburg-Landau functional
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.
- A. NACHMAN, University of Toronto, Toronto, Canada
Reconstructing Inhomogeneous Nonlinearities from Boundary Data
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
- JEAN-MICHEL ROQUEJOFFRE, CNRS-MIP and IUF, Université Paul Sabatier, Toulouse
Existence and stability of conical reaction-diffusion fronts
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.
- J. C. SAUT, Université Paris-Sud, 91405 Orsay
The global Cauchy problem for the Kadomtsev-Petviashvili I
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
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
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.
- ERIC SERE, Paris-Dauphine
A Hartree-Fock approximation of the polarized vacuum
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.
- TAI-PENG TSAI, University of British Columbia, Vancouver, BC V6T 1R9, Canada
Scattering for Gross-Pitaevskii equation
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.