


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 GinzburgLandau model of a superconducting sphere in a
uniform field

We consider the threedimensional GinzburgLandau model for a
spherical superconductor in a uniform applied field, in the limit as
the GinzburgLandau 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 GinzburgLandau 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 GinzburgLandau 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 JacquesLouis 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
problems

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 BoseEinstein condensate

We consider a twodimensional model for a rotating BoseEinstein
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 ThomasFermi 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.
 PIERRE DEGOND, MIP, CNRS and Université Paul Sabatier
Quantum hydrodynamics and diffusion models derived from the
entropy principle

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 equationofstate
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 nonlocal 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 wellestablished driftdiffusion and energytransport
models.
 NASSIF GHOUSSOUB, UBC
The optimal evolution of the free energy of interacting gases
and its applications

By studying the evolution of the freeinternal, potential and
interactiveenergies 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 FokkerPlanck 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 GinzburgLandau functional

The Jacobian estimates mentioned in the title of this talk are
estimates that control the Jacobian of a (typically complexvalued)
function in certain negative Sobolev norms by its GinzburgLandau
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
DirichlettoNeumann map on the boundary. This is joint work with
Victor Isakov.
 JEANMICHEL ROQUEJOFFRE, CNRSMIP and IUF, Université Paul Sabatier, Toulouse
Existence and stability of conical reactiondiffusion fronts

The premixed part of a Bunsen burner flame can be modelledin a very
crude approximationby a reactiondiffusion 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
(100yearold) 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
HaragusScheel (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 ParisSud, 91405 Orsay
The global Cauchy problem for the KadomtsevPetviashvili I
equation

The KadomstsevPetviahvili (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.
 ERIC SERE, ParisDauphine
A HartreeFock 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 D^{0} (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 BogoliubovDiracFock model (BDF). In this model,
the vacuum is represented by a bounded selfadjoint operator G
on L^{2} (R^{3}). 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 selfconsistent equation of HartreeFock
type. This minimizer is interpreted as the polarized Dirac sea.
This is joint work with Christian Hainzl and Mathieu Lewin.
 TAIPENG TSAI, University of British Columbia, Vancouver, BC V6T 1R9, Canada
Scattering for GrossPitaevskii equation

The GrossPitaevskii equation, a nonlinear Schroedinger equation with
nonzero boundary conditions, models superfluids and BoseEinstein
condensates. Recent mathematical work has focused on the shorttime
dynamics of vortex solutions, and existence of vortexpair traveling
waves. However, little seems to be known about the longtime behaviour
(eg. scattering theory, and the asymptotic stability of vortices). We
address the simplest such problemscattering around the vacuum
statewhich is already tricky due to the nonselfadjointness of
the linearized operator, and "longrange" nonlinearity. In
particular, our present methods are limited to higherdimension. This
is joint work in progress with S. Gustafson and K. Nakanishi.

