We study periodic minimizers of the Anisotropic Ginzburg-Landau and Lawrence-Doniach models for anisotropic superconductors, in various limiting regimes. We are particularly interested in determining the direction of the internal magnetic field (and vortex lattice) as a function of the applied external magnetic strength and its orientation with respect to the axes of anisotropy. We identify the corresponding lower critical fields, and compare the Lawrence-Doniach and anisotropic Ginzburg-Landau minimizers in the periodic setting.

This talk represents joint work with S. Alama and E. Sandier.

Nous étudions les minimiseurs périodiques du modèle de Ginzburg-Landau anisotrope ainsi que ceux du modèle de Lawrence-Doniach pour les supraconducteurs de hautes-températures anisotropes, et ce pour plusieurs régimes asymptotiques. Nous déterminons la direction du champs magnétique induit (et donc du réseau de vortex,) comme fonction de l'amplitude et de la direction du champs magnétique imposé. En particulier, nous trouvons le premier champs critique, et comparons les minimiseurs des modèles de Lawrence-Doniach et de Ginzburg-Landau anisotrope.

Ceci est un travail commun avec Stan Alama et Etienne Sandier.

Suppose that u(x,t) is a (possibly weak) solution of the Navier-Stokes equations on all of R³, or on the torus R³/ Z³. Let [^(u)](k,t) denote its Fourier transform. Then the energy spectrum of u(·,t) is the spherical integral

This is joint work with Andrei Biryuk.

I will first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space.

Then I will explain how these inequalities give us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by means of a blow-up method and a careful analysis of the convergence to a solution of a Liouville equation. In this way, the Onofri inequality appears as a limit case of the Caffarelli-Kohn-Nirenberg inequality.

This is joint work with J. Dolbeault and G. Tarantello

We analyze the nonlinear parabolic problem u_t = Du - [(lf(x))/((1+u)²)] on a bounded domain W of R^N with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at -1. When a voltage-represented here by l-is applied, the membrane deflects towards the ground plate and a snap-through may occur at finite "quenching time" T_l (W, f) when it exceeds a certain critical value l^* (W, f) (pull-in voltage). This creates a so-called "pull-in instability" which greatly affects the design of many devices. The challenge is to estimate l^* (W, f) and T_l (W, f) in terms of the shape (geometry) and the material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile f.

This is a joint work with Yujin Guo.

For the Gross-Pitaevskii (nonlinear Schroedinger) equation, which models superfluids (among other things), it is natural to consider non-zero boundary conditions at infinity. This results in richer dynamics than for the standard repulsive NLS with zero BCs at infinity-for example, traveling waves may form. On the other hand, we show that solutions with small, localized energy, disperse for large time, according to the linearized equation. Methods include certain quadratic transforms and bilinear estimates.

This is joint work with K. Nakanishi and T.-P. Tsai.

The first part of the talk is concerned with various generalizations of the usual notions of waves, fronts and propagation mean speed in a heterogeneous environment. The new notions involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. Transition waves not only extend the standard notions to a very general setting, but, under some appropriate assumptions in classical cases, they reduce to them. General intrinsic properties, monotonicity properties and uniqueness results for almost-planar fronts have been obtained with H. Berestycki.

New geometric situations can also be covered by the new notions. As an example, we will see in a second part (with H. Berestycki and H. Matano) how to describe the propagation of a generalized almost-planar front around an obstacle for bistable reaction-diffusion equations.

In this talk, I will report on recent results in collaboration with Frank Merle concerning the collision of two solitons for the subcritical gKdV equations ¶_t u + ¶_x ( ¶_x² u + g(u) ) = 0 on R. Under general assumptions on g(u), there exist stable solutions of the form u(t,x) = Q_c (x-c t), called solitons.

Considering 0 < c₂ < c₁, there exists a unique solution u(t) such that lim_{t® -¥} ||u(t) - Q_c1 (.- c₁ t) - Q_c2 (.- c₂ t)||_H¹(R) = 0. Formally, the solitons have to collide. We study this collision, i.e., the behavior of the solution u(t) for t bounded and for t® +¥, in the case where c₂ is small.

The first result concerns the global stability of the two soliton structure. Indeed, we prove that the two solitons survive the collision, with a possible residue and a possible change of sizes (and speeds) small with respect to Q_c2. Moreover, we show monotonicity properties: the size of the large soliton does not decrease and the size of the small soliton does not increase through the collision.

Second, we focus on the quartic case g(u)=u⁴, for which we give a sharp description of the collision. We prove that the residue is small but not zero. The collision is inelastic but very close to be elastic. This is in contrast with the integrable cases (g(u)=u² or u³) for which the collision is elastic (zero residue).

Finally, we exhibit special symmetric solutions which extend to the nonintegrable case the notion of 2-soliton for KdV.

This talk is concerned with the existence of traveling fronts for the equation:

The cubic nonlinear Schrödinger equation in dimension 2

This is joint work with Jim Colliander.

Reaction-diffusion of the KPP (Kolmogorov, Petrovskii, Piskunov) type have a semi-infinite interval of travelling wave velocities, and our goal is to examine how this family of waves attract the solutions of the evolution equation. We will mainly concentrate here on super-critical waves, i.e., those having a velocity larger than the smallest one. Well-known results (Uchiyama 1978, Bramson 1983) assert that an initial datum consisting of a super-critical wave, perturbed by a compactly-or rapidly decaying-function, will give rise to a solution converging to the initial wave, exponentially in time.

We wish to examine what happens when this assumption on the initial datum is relaxed. The scenario is the following: an initial datum trapped between two waves of the same velocity will evolve into a travelling wave profile, but with a local phase shift that may not converge to anything as time goes to infinity. In other words, convergence to a single wave does not survive.

We will describe this phenomenon, explain why it happens, and discuss some generalisations to models that are inhomogeneous in the space variable.

Joint work with M. Bages and P. Martinez.

Consider the motion of the ideal incompressible fluid on the surface of 2-d torus T² described by the Euler equations

Theorem    For any stationary solutions u₀, u₁ having equal energies and momenta, and for any e > 0 there exist T > 0 and a force f Î C^¥ (T×[0,T]) such that f transfers u₀ into u₁ during the time T, and

So, the flow may be controlled by a small (in L²) external force; most of the job is done by the fluid itself. One consequence of this fact is the absence of integrals of the Euler equations which are continuous in L² and different from the energy and momentum.

The proof is done by construction and uses the multiphase flow approximation for complex flows.