Journées Équations aux dérivées partielles
Forges-les-Eaux, 3-7 juin
GDR 2434 (CNRS)

Ses participants

Résumés / Abstracts

I. Serge Alinhac (Université Paris-Sud)

A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations.

The aim of this mini-course is twofold: describe quickly the framework of quasilinear wave equation with small data; and give a detailed sketch of the proofs of the blowup theorems in this framework. The first chapter introduces the main tools and concepts, and presents the main results as solutions of natural conjectures. The second chapter gives a self-contained account of geometric blowup and of its applications to present problem.

II. Jean-Marc Bouclet (University of Sussex)

Asymptotic behavior of regularized scattering phases for long range perturbations.

We define scattering phases for Schrodinger operators on $\mathbb{R}^d$ as limit of arguments of relative determinants. These phases can be defined for long range perturbations of the Laplacian; therefore they can replace the spectral shift function (SSF) of Birman-Krein's theory which can just be defined for some special short range perturbations (we shall recall this theory for non specialists). We prove the existence of asymptotic expansions for these phases, which generalize results on the SSF.

III. Lorenzo Brandolese (ÉNS de Cachan)

Localisation de la vorticité et applications au comportement asymptotique des solutions de Navier-Stokes.

We study the localization, with respect to the space, time and frequency variables, of the vorticity of a viscous flow filling the whole space. The special behavior of flows with highly oscillating vorticity is also discussed.

Nous présentons quelques résultats de localisation, en variables d'espace, temps et fréquence, pour la vorticité associée aux écoulements dans $\mathbb{R}^3$ d'un fluide visqueux. Nous étudions, ensuite, la localisation des écoulements caractérisés par des fortes oscillations de la vorticité.

IV. Alberto Bressan (S.I.S.S.A., Trieste)

On the Well Posedness of Vanishing Viscosity Limits.

This paper provides a survey of recent results concerning the stability and convergence of viscous approximations, for a strictly hyperbolic system of conservation laws in one space dimension. In the case of initial data with small total variation, the vanishing viscosity limit is well defined. It yields the unique entropy weak solution to the corresponding hyperbolic system.

V. Oleg Emanuvilov, Masahiro Yamamoto (Iowa State University)

Remarks on Carleman estimates and exact controllability of the Lamé system.

In this paper we established the Carleman estimate for the two dimensional Lamé system with the zero Dirichlet boundary conditions. Using this estimate we proved the exact controllability result for the Lamé system with with a control locally distributed over a subdomain which satisfies to a certain type of nontrapping conditions.

VI. Isabelle Gallagher, Dragos Iftimie, Fabrice Planchon (École Polytechnique)

Stabilité et asymptotique en temps grand de solutions globales des équations de Navier-Stokes.

We study a priori global strong solutions of the incompressible Navier-Stokes equations in three space dimensions. We prove that they behave for large times like small solutions, and in particular they decay to zero as time goes to infinity. Using that result, we prove a stability theorem showing that the set of initial data generating global solutions is open.

On étudie des solutions a priori globales des équations de Navier-Stokes incompressibles en trois dimensions d'espace. On montre qu'elles se comportent en grand temps comme des solutions petites, et en particulier elles décroissent vers zéro quand le temps tend vers l'infini. En utilisant ce résultat, on démontre que l'ensemble des données initiales générant des solutions globales est ouvert.

VII. Thierry Goudon, Pierre-Emmanuel Jabin, Alexis Vasseur (Université Nice-Sophia Antipolis)

Limites hydrodynamiques pour les équations de Vlasov-Stokes.

On présente quelques problèmes et résultats de type limites hydrodynamiques pour des modèles couplés fluide/cinétique décrivant l'interaction de particules avec un fluide en mouvement.

VIII. Frédéric Hérau (Université de Rennes 1)

Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential.

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak-Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications

IX. Thomas Kappeler (Universität Zürich)

Perturbations of the harmonic map equation.

We consider perturbations of the harmonic map equation in the case where the source and target manifolds are closed Riemannian manifolds and the latter is in addition of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. For generic perturbations the set of solutions is finite and we present a count of this set. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.

X. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao (University of Minnesota)

Existence globale et diffusion pour l'équation de Schrödinger nonlinéaire répulsive cubique sur $\mathbb{R}^3$ en dessous l'espace d'énergie.

We sketch a proof of global existence and scattering for the defocusing cubic nonlinear Schrödinger equation in $H^s(\mathbb{R}^3)$ for $s > \frac{4}{5}$. The proof uses a new estimate of Morawetz type.

Nous profilons une demonstration de l'existence globale et diffusion pour l'équation de Schrödinger nonlinéaire répulsive cubique avec données à $H^s(\mathbb{R}^3)$ pour $s > \frac{4}{5}$. Le raisonnement utilise une estimation nouvelle de type de Morawetz. Nous détaillerons la demonstration ailleurs.

XI. Norbert Mauser (Wolfgang Pauli Institute)

(Semi)classical limits of Schrödinger-Poisson systems via Wigner transforms.

We deal with classical and ``semiclassical limits'' , i.e. vanishing Planck constant $\hbar \simeq \epsilon \to 0$, eventually combined with a homogenization limit of a crystal lattice, of a class of "weakly nonlinear" NLS. The Schrödinger-Poisson (S-P) system for the wave functions $\{\psi^\epsilon_j(t,x)\}$ is transformed to the Wigner-Poisson (W-P) equation for a ``phase space function'' $f^\epsilon(t,x,\xi)$, the Wigner function. The weak limit of $f^\epsilon(t,x,\xi)$, as $\epsilon$ tends to $0$, is called the ``Wigner measure'' $f(t,x,\xi)$ (also called "semiclassical measure" by P. Gérard). The mathematically rigorous classical limit from S-P to the Vlasov-Poisson (V-P) system has been solved first by P.L. Lions and T. Paul (1993) and, independently, by P.A. Markowich and N.J. Mauser (1993). There the case of the so called ``completely mixed state'', i.e. $j=1,2,\dots,\infty$, was considered where strong additional assumptions can be posed on the initial data. For the so called ``pure state'' case where only one (or a finite number) of wave functions $\{\psi^\epsilon_j(t,x)\}$ is considered, recently P. Zhang, Y. Zheng and N.J. Mauser (2002) have given the limit from S-P to V-P in one space dimension for a very weak class of measure valued solutions of V-P that are not unique. For the setting in a crystal, as it occurs in semiconductor modeling, we consider Schrödinger equations with an additional periodic potential. This allows for the use of the concept of ``energy bands'', Bloch decomposition of $L^2$ etc. On the level of the Wigner transform the Wigner function $f^\epsilon(t,x,\xi)$ is replaced by the ``Wigner series'' $f^\epsilon(t,x,k)$, where the ``kinetic variable'' $k$ lives on the torus (``Brioullin zone'') instead of the whole space. Recently P. Bechouche, N.J. Mauser and F. Poupaud (2001) have given the rigorous ``semiclassical'' limit from S-P in a crystal to the ``semiclassical equations'', i.e. the ``semiconductor V-P system'', with the assumption of the initial data to be concentrated in isolated bands.

XII. Frank Merle, Pierre Raphael (Université de Cergy-Pontoise)

Blow up Dynamic and Upper Bound on the Blow up Rate for critical nonlinear Schrödinger Equation.

We consider the critical nonlinear Schrödinger equation $iu_t=-\Delta u-\vert u\vert^{{4\over N}}u$ with initial condition $u(0,x)=u_0$ in dimension $N$. For $u_0\in H^1$, local existence in time of solutions on an interval $[0,T)$ is known, and there exists finite time blow up solutions, that is $u_0$ such that $\lim_{t\rightarrow T<+\infty}\vert u_x(t)\vert _{L^2}=+\infty$. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data known to lead to blow up, no general understanding of the blow up dynamic is known. At first, we propose in this paper a general setting to study and understand small in a certain sense blow up solutions. Blow up in finite time follows for the whole class of initial data in $H^1$ with strictly negative energy, and one is able to prove a control from above of the blow up rate below the one of the known explicit explosive solution, which has strictly positive energy.

XIII. Christophe Prieur, Jonathan de Halleux (Université Paris-Sud)

Stabilization of a 1-D tank modeled by the shallow water equations.

We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by the shallow water equations. By means of a Lyapunov approach, we deduce control laws to stabilize the fluid's state and the tank's position. Although global asymptotic stability is yet to be proved, we numerically simulate the system and observe the stabilization for different control situations.

Nous considérons un bac de fluide soumis à un déplacement longitudinal. Nous modélisons le mouvement du fluide par les équations de Saint-Venant dont les équations linéarisées ne sont pas stabilisables. À l'aide d'une approche Lyapunov, nous déduisons des lois de contrôles qui numériquement stabilisent l'état du fluide et du bac.

XIV. Tristan Rivière (ETH Zürich)

Parois et vortex en micromagnétisme.

Nous présenterons l'énergie libre modélisant les états (polarisations) des materiaux ferromagnetiques. Le problème variationnel associé contient de nombreux régimes asymptotiques dans lesquels "on voit" se former des défauts du type vortex, du type paroi (Bloch et Neel Walls) ou du type mixte paroi-vortex (Cross-Tie Walls). Le but de cet exposé est de présenter les travaux qui s'efforcent de donner une justification mathématiqueà la création de ces singularités. Nous décrirons l'insuffisance des méthodes classiques de l'analyse fonctionnelle linéaire à rendre compte de ces phénomènes de perte de régularité et introduirons une approche mettant en oeuvre des outils de la théorie de la mesure géometrique appliqués a l'analyse des équations aux dérivees partielles. L'analyse des défauts en micromagnétisme a suscité des questions théoriques sur les lois de conservation scalaires non-linéaires. Nous presenterons à cette occasion un résultat récent géneralisant le phénomène de régularisation de Lax-Oleinick pour les lois scalaire à non-linearité strictement convexe $${\partial u\over\partial t}+{\partial f(u)\over\partial x}=0$$ ($f''>0$), au cas où la distribution entropique de sauts n'est pas forcément de signe uniforme mais est une mesure signée quelconque : pour tout $S$ dans $C^2$, $${\partial S(u)\over \partial t}+ {\partial\Phi(u)\over\partial x}$$ est une mesure de Radon ($(S,\Phi)$ paire entropie flux : $\Phi'=S'f'$). Nous démontrons alors que dans ce cas, en dimension 1+1, pour une donnée initiale mesurable et bornée quelconque, les ondes de chocs sont contenues dans une union au plus dénombrable de courbes $C^1$.

XV. Sergiu Klainerman, Igor Rodnianski (Princeton University)

Regularity and geometric properties of solutions of the Einstein-Vacuum equations.

We review recent results concerning the study of rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. We develop new analytic methods based on Strichartz type inequalities which results in a gain of half a derivative relative to the classical result. Our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates. The latter allows us to take full advantage of the specific structure of the Einstein equations.

XVI. Frédéric Rousset (ÉNS de Lyon)

Viscous Limits for strong shocks of one-dimensional systems of conservation laws.

We consider a piecewise smooth solution of a one-dimensional hyperbolic system of conservation laws with a single noncharacteristic Lax shock. We show that it is a zero dissipation limit assuming that there exist linearly stable viscous profiles associated with the discontinuities. In particular, following the approach of Grenier and Rousset (2001), we replace the smallness condition obtained by energy methods in Goodman and Xin (1992) by a weaker spectral assumption.

On considère un sytème hyperbolique de lois de conservation monodimensionnel $u_t + f(u)_x=0$, et une solution continue par morceaux avec un seul choc de ce système. En supposant qu'en tout point de discontinuité, il existe un profil visqueux linéairement stable, on montre qu'il existe une solution du système avec viscosité $u_t + f(u) _x = \epsilon u_{xx}$ qui tend vers la solution discontinue dans $L^\infty ([0,T] L^1)$ lorsque la viscosité tend vers zéro.

Édition numérique :

©

Cellule MathDoc (UMS 5638, UJF/CNRS)

©

Laboratoire de Mathématiques (UMR 6629, Université de Nantes)

Édition imprimée : ISBN 2-86939-188-9. Diffusion Laboratoire de Mathématiques (UMR 6629, Université de Nantes)