Journées Équations aux dérivées partielles
Forges-les-Eaux, 2-6 juin 2003
GDR 2434 (CNRS)

Ses participants

Résumés / Abstracts

I. Valeria Banica (Université d'Orsay)

Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain.

We concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound from below the blow-up rate for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than $(T-t)^{-1}$, the expected one. Moreover, we state that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.

II. Jean-François Bony, Laurent Michel (CNRS, Université de Bordeaux)

Microlocalization of resonant states and estimates of the residue of the scattering amplitude.

We obtain some microlocal estimates of the resonant states associated to a resonance $z_{0}$ of an $h$-differential operator. More precisely, we show that the normalized resonant states are ${\cal O} (\sqrt{ \vert{\rm Im} \, z_{0}\vert /h}$ $+ h^{\infty})$ outside the set of trapped trajectories and are ${\cal O} (h^{\infty})$ in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove such bound in some examples.

III. Rémi Carles (CNRS, Université de Bordeaux)

Changing blow-up time in nonlinear Schrödinger equations.

Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^2$-critical. On the other hand, introducing a ``repulsive'' harmonic potential prevents finite time blow-up, provided that this potential is sufficiently ``strong''. For the $L^2$-critical nonlinearity, this mechanism is explicit: according to the strength of the potential, blow-up is first delayed, then prevented.

IV. Walter Craig, Ana-Maria Matei (Mc Masters, Canada et Université Paris 9)

Sur la régularité des ondes progressives à la surface de l'eau.

Il a été établi par H. Lewy (1952) qu'une surface libre hydrodynamique qui est au moins $C^1$ dans un voisinage d'un point $q$ à la surface libre, est automatiquement $C^\omega$, éventuellement dans un voisinage plus petit de $q$. Ce résultat local est un exemple qui précedait la théorie dévélopée par D. Kinderlehrer, L. Nirenberg et J. Spruck (1977 - 79) demontrant que dans beaucoup de cas, des surfaces libres ne peuvent pas être d'une régularité arbitraire, et en particulier ils existent $m, \alpha$ telsque, si la surface en question est $C^{m,\alpha}$, alors automatiquement elle est $C^\omega$. Je vais exposer sur leurs methodes de transformation de Legendre/hodographe partielle, et des prolongements des methodes aux problemes en plusieures dimensions et avec la tension superficielle.

V. Nils Dencker (Université de Lund)

The proof of the Nirenberg-Treves conjecture.

We prove the Nirenberg-Treves conjecture: that for principal type pseudo-differential operators local solvability is equivalent to condition  ($\Psi$). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus. This makes it possible to reduce to the case when the gradient of the imaginary part is non-vanishing, and then the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the change of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of this submanifold. By using condition  ($\Psi$) and this weight, we can construct a multiplier which gives the estimate.

VI. David Dos Santos Ferreira (Université de Madrid)

Sharp ${L}^p$ Carleman estimates and unique continuation.

We will present a unique continuation result for solutions of second order differential equations of real principal type $P(x,D)u+V(x)u=0$ with critical potential $V$ in $L^{n/2}$ (where $n$ is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove $L^p$ Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its $L^p-L^{p'}$ boundedness properties.

VII. Christian Gérard (Université d'Orsay)

Théorie de la diffusion pour le modèle de Nelson et problème infrarouge.

We consider in this paper the scattering theory of infrared divergent massless Pauli-Fierz Hamiltonians. We show that the CCR representations obtained from the asymptotic field contain so-called coherent sectors describing an infinite number of asymptotically free bosons. We formulate some conjectures leading to mathematically well defined notion of inclusive and non-inclusive scattering cross-sections for Pauli-Fierz Hamiltonians. Finally we give a general description of the scattering theory of QFT models in the presence of coherent sectors for the asymptotic CCR representations.

Nous considérons dans cet exposé la théorie de la diffusion pour des modèles de Pauli-Fierz sans masse divergents infrarouge. Nous montrons que les représentations CCR obtenues a partir des champs asymptotiques contiennent des secteurs cohérents décrivant un nombre infini de bosons asymptotiquement libres. Nous formulons quelques conjectures qui conduisent a une notion bien définie de sections efficaces inclusives et non inclusives pour les Hamiltoniens de Pauli-Fierz. Finalement nous donnons une description générale de la théorie de la diffusion pour des modèles de théorie quantique des champs en présence de secteurs cohérents pour les représentations CCR asymptotiques.

VIII. David Gérard-Varet (ENS Lyon)

Convergence of the Rotating Fluids system in a domain with rough boundaries..

We consider a rotating fluid in a domain with rough horizontal boundaries. The Rossby number, kinematic viscosity and roughness are supposed of characteristic size $\epsilon$. We prove a convergence theorem on solutions of Navier-Stokes Coriolis equations, as $\epsilon$ goes to zero, in the well prepared case. We show in particular that the limit system is a two-dimensional Euler equation with a nonlinear damping term due to boundary layers. We thus generalize the results obtained on flat boundaries with the classical Ekman layers.

IX. François Golse (Université Paris 7)

The Mean-Field Limit for the Dynamics of Large Particle Systems.

This short course explains how the usual mean-field evolution PDEs in Statistical Physics -- such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations -- are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems.

X. Nader Masmoudi (Courant Institute)

Uniqueness results for some PDEs.

Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis Lions, with Kenji Nakanishi and with Fabrice Planchon.

XI. Jonathan C. Mattingly (Université de Duke)

On Recent Progress for the Stochastic Navier Stokes Equations.

We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. A few general theorems are given. Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and hypoellipticity are all discussed.

XII. Francine Meylan, Nordine Mir, Dmitri Zaitsev (Université de Rouen)

On some rigidity properties of mappings between CR-submanifolds in complex space.

We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.

XIII. Marius Paicu (Université d'Évry et École Polytechnique)

Fluides incompressibles horizontalement visqueux.

Motivé par l'étude des fluides tournants entre deux plaques, nous considérons l'équation tridimensionnelle de Navier-Stokes incompressible avec viscosité verticale nulle. Nous démontrons l'existence locale et l'unicité de la solution dans un espace critique (invariant par le changement d'échelle de l'équation). La solution est globale en temps si la donnée initiale est petite par rapport à la viscosité horizontale. Nous obtenons l'unicité de la solution dans un espace plus grand que l'espace des données pour lesquelles on sait résoudre l'équation. La démonstration s'appuie sur un découpage adapté en fréquences verticales (on estime différemment la partie "basses fréquences" et la partie "hautes fréquences") et sur le contrôle précis de la régularité dans les variables horizontales.

XIV. Luis Vega (Université del Pais Vasco)

Kink solutions of the binormal flow.

I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow: $X_{t}=X_{s}\times X_{ss}$ which preserve the length parametrization. Above $X(s,t)$ is a curve in $\mathbb{R}^3$, $s\in\mathbb{R}$ the arclength parameter, and $t$ denote the temporal variable. This flow appeared for the first time in the work of Da Rios (1906) as a crude approximation to the evolution of a vortex filament under Euler equation, and it is intimately related to the focusing cubic nonlinear Schrödinger equation through the so called Hasimoto transformation. These solutions show the formation of singularities in finite time in the shape of either just a kink (zero angular momentum) or a kink together with a logarithmic correction in the shape of a spiral (non trivial angular momentum).

XV. Steve Zelditch (John Hopkins University)

Billiards and boundary traces of eigenfunctions.

This is a report on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.

Édition numérique :

©

Cellule MathDoc (UMS 5638, UJF/CNRS)

©

Laboratoire de Mathématiques (UMR 6629, Université de Nantes)

Édition imprimée : ISBN 2-86939-207-9. Diffusion Laboratoire de Mathématiques (UMR 6629, Université de Nantes)