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au calcul scientifique

à l'occasion de son soixantième anniversaire

Paris, 2-6 juillet 2007

Structure of null sets in Euclidean spaces: results and open problemsI will describe a decomposition theorem for sets of measure zero in the plane which can be deduced from an elementary combinatorial result (Dillworth's lemma). I will then outline a few applications to elementary questions in Geometric Measure Theory. The extension of these results to higher dimensions is mostly open. This is joint work with D. Preiss (Warwick) and M. Csörnyei (University College London). Contact homology via Legendrian curves: definition and
first steps of computationUsing the Legendrian curves associated to a "dual" form, we have defined in an earlier work a contact homology for three dimensional contact structures. We will establish in this talk that this definition holds under full generality. We will show that compactness holds in this framework and we will provide the value of this homology for odd generators. On the field-matter interaction in electromagnetism:
a weak convergence approachIt has been a long standing goal of theoretical physics to derive interaction between field and matter directly from field equations, without any coupling, matter being interpreted as singularities of the field (such as vortices, for instance) or, alternately, as zones of high intensity of the field (when the field equations do not generate singularities). In the case of classical electromagnetism, this idea has been introduced by Gustav Mie, then followed by Max Born and many others. Starting from the nonlinear theory of Born and Infeld, we will see how the field-matter interaction follows from the original field equations (the Born-Infeld system) after performing two steps: i) adding the conservation of energy-momentum equations to the original field equations, and ii) introducing the weak closure of the algebraic manifold in which the solutions take their value. We will also discuss, with a quite different point of view, a concept of viscosity solutions in the large for this kind of nonlinear systems. Fully nonlinear equations for integral diffusionsWe will present the elements of a regularity theory (ABP theorem, estimates, Harnack type inequality) for "bounded measurable" and "fully nonlinear" diffusions of fractional type. This is a joint project with Luis Silvestre. Hyperbolic balance laws with dissipationThe lecture will characterize source terms of hyperbolic systems of balance laws that induce dissipation, in the context of BV solutions, and will discuss implications to the theory of relaxation. Hardy-Dirac inequalities and applicationsIn this talk I will present various versions of Hardy-like inequalities for the Dirac operator. Particular attention will be given to the case of multipolar and magnetic potentials, where the physical model behind the inequalities as well as their implications will be described in details. A nonlinear partial differential equation model for lakes and riversI will discuss a simple fast diffusion partial differential equation that predicts the formation of "lakes" and "rivers" on a given landscape. The sharp Sobolev inequality in quantitative formWe present a quantitative version of the sharp Sobolev inequality in , 1<p<n
, with a remainder term involving the distance
from the functions for which the equality is attained.
Wave equation, Strichartz inequality and Lorentz transformationsLet u be a solution of the three-dimensional wave equation.
The Strichartz inequality
is an important tool for studying nonlinear perturbations of the wave equation. The goal of this talk is to describe, for a given sequence of such solutions u with Cauchy data
which converge weakly to (0,0) in
, the obstruction to
the convergence to 0
of the
norm of u,
and in particular the role played by
Lorentz transformations in this obstruction. Applications to cubic
perturbations of the wave equation will be given.
Numerical simulation of a channel flow with chaotically moving boundaryIn this talk I will present joint work with R. Azencott and G. Guidoboni. One of the objectives of our resarch is to simulate the flow of incompressible viscous fluids modelled by the Navier-Stokes equations when (a part of) the boundary is moving randomly. I will present here a first step in this direction. We consider a pressure driven flow in a two-dimensional channel when the two walls parallel to the pressure gradient have a chaotic motion preserving the area of the channel. In order to solve the above problem numerically we combine a time discretization by operator-splitting with a geometrical transformation mapping the flow domain into a fixed rectangle, allowing thus a space discretization by a finite element method taking advantage of a fixed triangulation. The results of numerical experiments suggest a resonance phenomenon when (some kind of) a fundamental frequency varies; this implies in particular that the variation of the equivalent viscosity of the time averaged flow is not monotonic. The weak limit of semigroupsThe weak limit of a sequence of strongly continuous semigroups needs not be a semigroup. The zero dispersion limit of the KdV equation is an example, and possibly the description of a turbulent flow. Mean field gamesIn this talk we derive new systems of equations by taking the limits of Nash equilibria as the number of players goes to infinity. We indicate some aspects of the rigorous derivation of such systems and discuss their mathematical structure. We shall also mention the connections with the Hartree type systems in Quantum Mechanics, the Euler equations, the optimal control of partial differential equations... Existence results for micro-macro models of polymeric flowsSystems coupling fluids and polymers are of great interest in many branches of applied physics, chemistry and biology. Although a polymer molecule may be a very complicated object, there are simple theories to model it. One of these models is the FENE (Finite Extensible Nonlinear Elastic) dumbbell model. Another model is the Doi model (or Rigid model) where each polymer is modeled by a vector on the sphere. We will present some local and global existence results for these models. New estimates on singular integral operatorsAn improvement on the T(1) theorem by David and Journé
will be discussed. This research is a joint work with Professor
Qixiang Yang.
Modifications to Newton's second law and continuum
elastodynamicsNewton's law, F = m a, has withstood centuries but when applied
to a body's motion in principle requires tracking trillions of
electrons and nuclei. Instead one "averages things out" and applies
Newton's law macroscopically. However, building upon previous work, we
show that if one does this, the macroscopic version of Newton's law
takes a different form in which force depends on acceleration not just
at the present time but also at previous times. We provide simple
models with a complex and direction dependent inertia at a given
frequency. The elastodynamic equations are also modified, and we
obtain new equations, generalizing those found by Willis in the
1980's, which govern the behavior of the ensemble averaged weighted
displacement field. The weighting function may be zero in regions
where there are voids, or where the displacement field is
unobservable. This research is joint work with John Willis.
Extensions of a simple inequalityThe inequality refers to a nonnegative function u defined on
(-1,1). It estimates |du/dt(0)| in terms of
u(0) and of the maximum of the second derivative of u.
Various extensions are given, including some to higher dimensions.
Numerical challenges in mathematical financeSince the work of Black and Scholes, the modelling of options and other derivatives has become quite sophisticated. Analytical solutions of the models do not exist in most cases and numerical simulations are necessary. Some of the models require the solving of partial differential equations in many dimensions and are numerically very expensive. In this talk we shall review these topics and present personal contributions for the direct computation of European and American basket options, for stochastic volatility models and for their calibration. For the calibration of local volatilities in the Black-Scholes model, we shall review Dupire's approach and show that duality can be used at the discrete level for any linear model. Numerical results will be presented when spline approximations are used to reduce the number of parameters. Analysis aspects of Willmore surfacesWe present a new formulation of the Euler-Lagrange equation of the Willmore functional for immersed surfaces in . This new formulation of Willmore equation is in divergence form. Moreover the nonlinearities are made of jacobians. Additionally, if denotes the mean curvature vector of the surface, this new form reads as , where is a well defined locally invertible elliptic self-adjoint operator. These three facts have numerous consequences in the analysis of Willmore surfaces. A first consequence is that the long standing open problem of giving a meaning to the Willmore Euler-Lagrange equation for immersions having only bounded second fundamental form is now solved. We then establish the regularity of weak Willmore immersions with bounded second fundamental form. The proof of this result is based on the discoveries of conservation laws for Willmore immersions and of grad-curl structures which are preserved under weak convergences. We establish then a weak compactness result for Willmore surfaces of energy less than (the Li-Yau condition which ensures the embeddedness of the surface). This theorem is based on a point removability result that we prove for Wilmore surfaces in . Finally, we deduce from this point removability result the strong compactness, modulo the Möbius group action, of Willmore tori below the energy level in dimensions 3 and 4. Heuristic and rigorous treatment of ill-posed singular
perturbation problemsIn this lecture, I will present joint work with Yu. V. Egorov and N. Meunier. We consider singularly perturbed elliptic boundary value problems depending on a parameter . The problem under consideration is classical for but is ill-posed for since its boundary condition does not satisfy the Shapiro-Lopatinskii condition on some part of the boundary. We use a non variational framework due to Caillerie, which allows one to prove the existence of a unique limit in an appropriate abstract space. We consider more general domains (here the domains are two dimensional manifolds with boundary) than those considered in previous works on the subject. A heuristic simplifying argument leads to an equivalence between the problem for a general geometry and a simpler problem formulated on a one dimensional manifold. That kind of problem is motivated by the theory of thin shells. Initial-boundary value problems for wave-like systems with
Neumann-type boundary conditionWe consider linear second order hyperbolic initial-boundary value problems, where we assume that the partial differential equation is the Euler-Lagrange equation of some Lagrangian. The homogeneous boundary conditions are those coming from the variational formulation, when the variations take arbitrary values at the boundary. The simplest example is that of the wave equation with Neumann boundary condition. Classical is also the system of linear elasticity with zero normal stress at the boundary. We show that such an initial-boundary value problem cannot satisfy the uniform Kreiss-Lopatinskii condition. Therefore the corresponding non-homogeneous initial boundary value problem is not strongly stable in Kreiss' sense. But since we are concerned by the homogeneous boundary condition, we give a new definition of strong stability that is appropriate for nonlinear iterations. We characterize this stability condition in various terms: coercivity of the internal energy, non-vanishing of the Lopatinskii determinant on an appropriate set, positivity of some parametrized Hermitian matrix. The theory gives quite comfortable results. In particular, the passage from the constant coefficients case in a half-space to variable coefficients in a bounded domain is quite simple and does not require pseudo-differential tools. At last, we prove that such problems display surface waves in every direction along the boundary. Acknowledgement: I began this research after a discussion with Luc, although I never answered his precise question. Heat flow versus Schrödinger flow for the Ginzburg-Landau energy
in two dimensionsThe Gross-Pitaevskii and the parabolic Ginzburg-Landau equations correspond respectively to the Schrödinger and heat flows for the Ginzburg-Landau energy functional. Among the interesting features of these two systems is the fact that different regimes lead to a rich variety of limit systems: in the planar case, these include for example vortex motions, traveling waves, and phase oscillations governed by the linear heat and wave equations. In this talk I will review some recent works aimed at understanding each of these modes as well as their interactions. On the stability of a uniformly rotating viscous incompressible self-gravitating fluidWe give a justification of the principle of minimization of the potential energy for the stability problem of a uniformly rotating viscous incompressible self-gravitating fluid. Special attention is given to the case where the surface tension is not taken into account. We prove that the regime of rigid rotation is stable when the second variation of the potential energy is positive definite, and unstable when it can take negative values. The proof is based on the analysis of a free boundary problem for perturbations of the velocity and of the pressure of the rotating fluid, and on the analysis of the corresponding linearized problem. Backward stochastic differential equations and
nonlinear partial differential equationsBackward stochastic differential equations are a system of stochastic equations with both initial and terminal data. These equations appear naturally in optimal stochastic control and in mathematical finance, and they have been studied extensively in the last decade. There is also an interesting connection with nonlinear parabolic equations through a simple use of stochastic calculus. In this talk, I will outline this connection and use it to develop a Monte-Carlo type numerical scheme for nonlinear partial differential equations. On Landau solutions of the Navier-Stokes equationsIn 1944 L.D. Landau calculated a very interesting family of explicit solutions of the steady-state 3d Navier-Stokes equations. The solutions are derived under certain assumptions of symmetry, which reduce the Navier-Stokes equations to a system of ordinary differential equations. We investigate what happens when some of the symmetry conditions are dropped (and we have to deal with a system of partial differential equations). Possible implications of these calculations for more general classes of solutions will also be discussed. |