Programme

Semaine Lun. 08 Mar. 09 Mer. 10 Jeu. 11 Ven. 12 Liste

	Lun. 08	Mar. 09	Mer. 10	Jeu. 11	Ven. 12
09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00	Pause café 10:00 - 10:30 (30min) Pause café Valeria Banica 10:30 - 11:10 (40min) Valeria Banica Title: On the binormal flow for infinite curves. Abstract: The binormal flow is a geometric flow of curves in R^3 that models vortex filament dynamics in fluids. We construct solutions for infinite curves data including the known examples of solutions that generate singularities in finite time. In this general low regularity context we use geometric measure theory tools. More precisely, we introduce a notion of renormalized length for infinite curves and work in the framework of locally integral currents. This is a joint work with Bob Jerrard and Didier Smets. Kenji Nakanishi 11:20 - 12:00 (40min) Kenji Nakanishi Title: Scattering for the 4D Zakharov system below the ground states. Abstract: This is joint work with Timothy Candy. For the Zakharov system in four space dimensions, we prove that all solutions inside the potential well of the ground states are scattering in the energy space. It is an extension from the radial case and the proof in the non-radial case was already reduced by Candy to exclusion of a minimal non-scattering solution. The main difficulty, compared with the radial case and with the nonlinear Schrodinger or wave equation, comes from absence of a center moving by the momentum and the Galilei/Lorentz invariance. It is resolved by combination of two distinct arguments depending on the motion of center, subsonic or supersonic. The former is a virial-variational argument for the Zakharov system, and the latter is a space-time estimate for the linear wave equation. Déjeuner 12:10 - 14:00 (1h50) Déjeuner Nikolay Tzvetkov 15:00 - 15:40 (40min) Nikolay Tzvetkov Title: On the statistical description of the flow of non linear dispersive PDE. Abstract: We study the three dimensional cubic defocusing wave equation with Gaussian initial data. We show that the law of the solution is quasi-Gaussian in low regularity regimes exploiting renormalised energies, normal form reductions, remarkable cancellations, non commutative Khinchin inequalities and cut-offs designed according to the probabilistic well-posedness theory developed by Nicolas Burq and the speaker around 2010. This is a joint work with Chenmin Sun and Leonardo Tolomeo. Isabelle Gallagher 15:50 - 16:30 (40min) Isabelle Gallagher Title: Fourier analysis on the Heisenberg and Engel groups, and applications. Abstract: The Heisenberg and Engel groups are prototypes of Carnot groups of steps 2 and 3 respectively. We shall present properties of the sublaplacian on these groups, which are related via the Fourier transform to the quadratic and quartic oscillators. The spectral analysis of these operators leads to estimates on the convolution kernel of operators of the type $F(-\Delta)$, which in turn allow to recover classical functional embeddings, and estimates on the Schrödinger propagator for instance, via Fourier techniques. This corresponds to joint works with Hajer Bahouri, Davide Barilari and Matthieu Léautaud. Pause café 16:30 - 17:00 (30min) Pause café Thomas Duyckaerts 17:00 - 17:40 (40min) Thomas Duyckaerts Title: A journey through the work of Nicolas Burq	Thomas Alazard 9:30 - 10:10 (40min) Thomas Alazard Title: A fluid-structure problem with free surface. Abstract: We consider a fluid-structure problem governed by the incompressible Euler equations with a free surface, in a domain whose interface is governed by linear elasticity. Our main result asserts that the Cauchy problem is globally well-posed in time for any irrotational initial data in the energy space, without smallness assumption. In the absence of parabolic regularization, we prove that the system may be reformulated as a nonlinear Schrödinger-type equation. This enables the construction of solutions that are highly irregular from the perspective of inviscid fluid dynamics: the initial fluid velocity possesses merely half a derivative in L2. Joint work with Chengyang Shao (IHES) and Haocheng Yang (NYU Abu Dhabi). Claude Zuily 10:20 - 11:00 (40min) Claude Zuily Unique continuation for fractional powers oh sub-elliptic operators Pause café 11:00 - 11:30 (30min) Pause café Sebastian Herr 11:30 - 12:10 (40min) Sebastian Herr Title: The cubic Dirac equation and its non-relativistic limit. Abstract: A scale-invariant global well-posedness and scattering theory for cubic Dirac equations will be presented. It is based on a modular approach using critical adapted function spaces and bilinear Fourier restriction theory. As an application, we can study the non-relativistic limit, more precisely, convergence towards a system of cubic nonlinear Schrödinger equations. This is joint work with Timothy Candy. Déjeuner 12:10 - 14:00 (1h50) Déjeuner Jérôme Le Rousseau 15:00 - 15:40 (40min) Jérôme Le Rousseau Title: Observation of the Stokes system under general boundary conditions. Abstract. Near a boundary, we recast the Stokes system as a first-order system. A microlocal matrix reduction then makes it possible to derive a Carleman estimate. Because of the presence of Jordan blocks, this estimate exhibits a loss of a full derivative, which makes its derivation delicate, in particular when patching together microlocal estimates. The boundary estimate can then be used to obtain an observability estimate for the Stokes or Oseen system, the latter being a linearization of the Navier–Stokes system. Controllability results can then be deduced. General boundary conditions are considered, in the spirit of the Lopatinskii–Shapiro conditions, including the classical Dirichlet, Navier, and Neumann conditions. This joint work with Luc Robbiano. Belhassen Dehman 15:50 - 16:30 (40min) Belhassen Dehman Title: Regional and partial observability and control of waves. Abstract: We establish sharp regional observability results for solutions of the wave equation in a bounded domain Ω of R^n. Given a non-empty open subset ω ⊂ Ω, we derive estimates for the energy of initial data supported in another open subset O ⊂ Ω, in terms of the energy measured on the observation set (0,T )×ω. This holds under a suitable geometric condition relating the time horizon T and the pair of subdomains (ω,O), which is weaker than the so-called Geometric Control Condition (GCC). A notable feature of our approach is that it remains effective in settings where classical unique continuation results doesn’t apply. The proof combines a high-frequency observability estimate — based on the propagation of singularities — with a compactness-uniqueness argument that exploits the unique continuation properties of elliptic operators. By duality, this observability result leads to regional controllability results. Pause café 16:30 - 17:00 (30min) Pause café Gigliola Staffilani 17:00 - 17:40 (40min) Gigliola Staffilani Title: Some mathematically rigorous results in wave turbulence theory. Abstract: In this talk we give an overview of some old and new mathematical advances in wave turbulence theory. We start with the original perspective of Bourgain on the study of the energy spectrum for a periodic nonlinear Schrödinger equation via growth of Sobolev norms. Then we move to the wave kinetic equations as effective equations for the energy spectrum and we give examples of rigorous proofs of condensate growth and energy transfer.	Herbert Koch 9:30 - 10:10 (40min) Herbert Koch Title: Smoothing for the scattering operator for gKdV and NLS. Abstract: Consider a nonlinear dispersive equation (defocusing or small data) like gKdV or NLS in the mass critical or supercritical regime. The inverse of the wave operator Omega_+ maps the initial data u_0 to the forward scattering data v_0 so that u(t)-v(t) converges to zero. Here u(t) is the solution to the nonlinear dispersive equation and v(t) is the solution to the linear dispersive equation with initial data u_0 resp. v_0. Above the scaling critical space u_0-v_0 is more regular than u_0. A similar phenomenon occurs for stochastic initial data. This is joint work with Nicolas Burq, Nikolay Tzvetkov and Nicola Visciglia. Pierre Raphaël 10:20 - 11:00 (40min) Pierre Raphaël Title: On the discovery of defocusing blow up bubbles. Abstract: I will describe the 20 years long research path which led to the discovery of defocusing super critical blow up bubbles and their viscous fluid analogues, and how spectral analysis played a fundamental role. Pause café 11:00 - 11:30 (30min) Pause café Oana Ivanovici 11:30 - 12:10 (40min) Oana Ivanovici Déjeuner 12:10 - 14:00 (1h50) Déjeuner	Leonardo Tolomeo 9:30 - 10:10 (40min) Leonardo Tolomeo Title: A statistical consequence of the soliton resolution conjecture for the focusing nonlinear Schrödinger equation. Abstract: In this talk, I will discuss concentration of measure phenomena for the Gibbs of the focusing, mass-subcritical Schrödinger equation on the real line. I will first discuss how these results connect to the soliton resolution conjecture for NLS, and then explain how to approach this family of problems, combining the variational representation of solitons with more probabilistic techniques. This talk is based on joint works with T. Oh (Edinburgh), M. Okamoto (Osaka), H. Weber (Münster), J. Forlano (Monash), and F. Höfer (Münster). Fabrice Planchon 10:20 - 11:00 (40min) Fabrice Planchon Title: 10 ways to prove local smoothing. Abstract: I will revisit our old results (joint with N. Burq) on the 1d Schrödinger equation with a BV coefficient, with emphasis on how to avoid spectral theory (to the extend it is possible to do so!) and why we should. In the process we will greatly simplify the original proofs and discuss possible extensions. Pause café 11:00 - 11:30 (30min) Pause café Wilhelm Schlag 11:30 - 12:10 (40min) Wilhelm Schlag Title: Radiation Damping and Asymptotic Stability of the Degree-One Vortex in the Abelian Yang-Mills-Higgs Model. Abstract: The abelian Yang-Mills-Higgs equations in two space dimensions admit topological vortex solutions. I will discuss the asymptotic stability of the degree-one vortex at self-dual coupling, for small corotational perturbations in Stuart’s gauge.Joint work with Jose Palacios, Fabio Pusateri, Jonas Lührmann, and Sohrab Shahshahani. Déjeuner 12:10 - 14:00 (1h50) Déjeuner Maciej Zworski 15:00 - 15:40 (40min) Maciej Zworski Title: WKB structure in a scalar model of flat bands Abstract: The scalar model of flat bands is a simplification of models in condensed matter physics. It allows the study of relevant spectral problems using a 2nd order scalar equation, akin to the Schroedinger equation with the square of dbar on a torus replacing the Laplacian. It displays many features of original models such as the ``quantisation" of the reciprocals of magic angles at which flat bands appear. The space of solutions can be described using a rank 2 holomorphic vector bundle over the torus and its properties as alpha varies are related to the structure of bands leading to a trichotomy: tangential touching (most of alphas), Dirac points (discrete set of alphas) and flat bands (discrete set). (Mengxuan Yang and Bryan Li observed that the same argument works in a more physically realistic setting of twisted two-layered wafers of graphene.) In my talk I will describe the basic properties of the scalar model and of the ge Chenmin Sun 15:50 - 16:30 (40min) Chenmin Sun Title: Solving NLS on the three-dimensional Ball with radial, supercritical random data. Abstract: We consider the cubic nonlinear Schrödinger equation on the three-dimensional ball with radial random initial data in a supercritical regime. We construct probabilistic strong solutions, substantially improving a previous result of Bourgain and Bulut. The criticality arises from a divergent contribution that, unlike for the NLS on the torus, cannot be removed by a global transformation. We overcome this obstruction by introducing a refined gauge transform which leaves the equation unchanged. I will then explain how this gauge transform can be incorporated into the random averaging operator ansatz to solve the problem. This is joint work with Nicolas Burq, Nicolas Camps, and Nikolay Tzvetkov. Pause café 16:30 - 17:00 (30min) Pause café Massimiliano Berti   17:00 - 17:40 (40min) Massimiliano Berti   Title: Long time dynamics of space periodic water waves. Abstract: The goal of this talk is to review recent advances regarding the long-time dynamics of space-periodic water waves, focusing on 1) bifurcation of quasi-periodic solutions, both standing and traveling; 2) long-time well-posedness results; 3) modulational instability of Stokes waves. These results rely on unconventional approaches to KAM and Birkhoff normal form theories for Hamiltonian quasi-linear PDEs and symplectic Kato perturbation theory for separated eigenvalues of reversible and Hamiltonian operators.	Luigi Ambrosio 9:30 - 10:10 (40min) Luigi Ambrosio Title: Well posedness of ODE's for nonsmooth velocities and in non Euclidean ambient spaces: a survey. Abstract: In this lecture I will make a survey of the theory of regular lagrangian flows, that allows to prove existence, uniqueness and stability for ODE's associated to vector fields with little regularity, beyond the classical Cauchy-Lipschitz theory. Initiated in a minal work by Di Perna and Lions who dealt with Sobolev vector fields, the RLF axiomatization provides natural links with Probability. More recent developments, including the regularity of the flow map, counterexamples and the case when even the ambient space is not smooth, will be discussed. Daniel Tataru 10:20 - 11:00 (40min) Daniel Tataru Title: Global solutions for 3D gravity water waves. Abstract: The aim of this talk is to present work in progress on the problem of local and global well-posedness for gravity water waves in the small-data, low-regularity regime, for fluids of infinite depth and infinite extent in spatial dimensions n≥3. More specifically, our ongoing work seeks to simultaneously achieve substantial improvements in both the local and global theory for these equations, reaching nearly optimal Sobolev regularity thresholds. On the local side, we obtain sharp improvements in the well-posedness theory for the gravity wave problem: up to scaling regularity in dimensions n≥4, and within 1/12 of a derivative above scaling in dimension n=3.On the global side, we establish global well-posedness for small initial data in a critical Besov space, without requiring spatial localization assumptions. These results fit within the framework of the “Nonlocalized Data Global Well-posedness Conjecture” , jointly propose Pause café 11:00 - 11:30 (30min) Pause café Patrick Gérard 11:30 - 12:10 (40min) Patrick Gérard Title: Soliton and breather resolution for the cubic Szegö flow on the line. Abstract: We investigate the long--time behaviour of the solutions of the cubic Szegö equation on the line in the Sobolev space $H^{1/2}(\R)$. We prove that, for every datum of which the Lax operator has simple positive spectrum, the solution asymptotically decouples as an infinite sum of traveling quasi-periodic breather solutions. Under an additional generic condition on the data, we prove that these traveling breather solutions are in fact soliton solutions, leading to a soliton resolution theorem. This talk is based on a recent joint work with Sandrine Grellier. Déjeuner 12:10 - 14:00 (1h50) Déjeuner

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