Abstracts

  • Valeria Banica - Wavy blow-up for the 1D cubic NLS

    In this talk I will display a class of wavy initial data that lead to blow-up in finite time via the classical 1D cubic Schroedinger equation. This is a joint work with Renato Luca, Nikolay Tzvetkov and Luis Vega.

  • Nabile Boussaid - Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schrödinger operators

    In this talk I will present a joint work with Andrew Comech (Texas A&M). Our primary interest is on the limiting absorption principle. Such a tool is useful in spectral analysis. It is also used to obtain dispersive estimates and then to analyze long time evolutions of associated nonlinear problems. In dimension 1 and 2, in contrast with higher dimensions, the free linear Schrodinger operator has no limiting absorption principle. This makes all this classical perturbative approach much more involved. The absence of limiting absorption principle (LAP) in the vicinity of some point is equivalent to the presence of a virtual level at this point. But it is also known that virtual levels are unstable by perturbations leading to bifurcation of eigensates. Our work is an attempt to understand the different characterizations of virtual levels and to provide limiting absorption principle for small perturbations of the free Schrödinger operators in dimension 1 and 2.

  • Piero D'ancona - Scattering for the NLS with variable coefficients on the line

    In recent years an efficient framework was established to prove scattering for nonlinear dispersive equations, based on the combination of concentration-compactness principles and induction on energy arguments. Originally developed by Kenig and Merle, the framework has been adapted to several equations with constant coefficients. The presence of potential perturbations or variable coefficients introduces new difficulties due to unisotropy. In this talk I shall report on some new results, obtained in collaboration with Angelo Zanni (Roma), concerning scattering for a defocusing, subcritical NLS in one space dimension, with fully variable coefficients.

  • Roland Donninger - Optimal blowup stability for wave maps

    I will present a recent result, obtained jointly with David Wallauch (Vienna), on the stability of blowup for supercritical wave maps. More precisely, we consider corotational wave maps on (1 + 4)-dimensional Minkowski space into the 4-sphere. We show that the explicit self-similar blowup is stable under perturbations that are small in H^2, which is the critical regularity. The main technical tool is a novel set of Strichartz estimates in similarity coordinates.

  • François Hamel - Spreading speeds and one-dimensional symmetry for reaction-diffusion equations in R^N.

    The talk will focus on the large-time dynamics of bounded solutions of reaction-diffusion equations in R^N with unbounded initial support. I will discuss the existence of spreading sets and Freidlin-Gartner formulas for the spreading speeds of the solutions in any direction, in connection with the existence of planar traveling waves. I will also explain some results on the asymptotic one-dimensional symmetry of the elements of the Omega-limit set of the solutions, in the spirit of a famous De Giorgi conjecture on solutions of some elliptic equations in R^N. The talk is based on joint works with Luca Rossi.

  • Sebastian Herr - Global wellposedness of the Zakharov System below the ground state

    The Zakharov system is a quadratically coupled system of a Schrödinger and a wave equation, which is related to the focussing cubic Schrödinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that it is globally well-posed in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schrödinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate. This is oint work with Timothy Candy and Kenji Nakanishi.

  • Renato Lucà - On the topology of the magnetic lines of solutions of MHD

    We give examples of solutions to the magnetohydrodynamic (MHD) equation with positive resistivity for which the topology of the magnetic lines changes under the flow. By Alfvén's theorem this is known to be impossible in the ideal case (resistivity = 0). This is a joint work with Pedro Caro and Gennaro Ciampa.

  • Alberto Maspero - Full description of Benjamin-Feir instability of Stokes waves in finite and deep water

    Small-amplitude, traveling, space periodic solutions – called Stokes waves – of the 2 dimensional gravity water waves equations in finite and deep water are linearly unstable with respect to longwave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure eight, parameterized by the Floquet exponent, in full agreement with numerical simulations. This is a joint work with M. Berti and P. Ventura.

  • Diego Noja - The NLS equation with a point interaction in two and three dimensions

    The NLS equation with a power nonlinearity and point interaction (a "delta potential") is well studied in dimension one but only recently the two and three dimensional case have been tackled. In the talk, the well posedness of the Cauchy problem will be treated and a discussion of some asymptotic properties will be given, in particular as regards the blow-up for strong nonlinearities and the absence of scattering for low nonlinearity. Work in collaboration with Claudio Cacciapuoti and Domenico Finco.

  • Julien Sabin - Strong coupling limit of the Dirac-Klein-Gordon system

    In this talk I will explain how to derive the nonlinear Dirac equation from the Dirac-Klein-Gordon system in the strong coupling limit. This is a joint work with Jonas Lampart, Loïc Le Treust, and Simona Rota Nodari.

  • Eric Séré - Nash-Moser without Newton

    The Nash-Moser theorem allows to solve a functional equation F(u)=0 in a "scale" of Banach spaces, assuming that F(0) is very small and that near 0 the differential DF has a right inverse which loses derivatives. The classical proof uses a Newton iteration scheme, which converges when F is of class C^2. In contrast, we only assume that F is continuous and has a Gâteau first differential, which is right-invertible with loss of derivatives. In our iteration scheme, each step consists in solving a Galerkin approximation of the equation, using Ekeland's variational principle. We apply our method to a singular perturbation problem with loss of derivatives studied by Texier-Zumbrun. We will compare the two results and we will show that ours improves significantly on theirs, when applied, in particular, to a nonlinear Schrodinger Cauchy problem with highly oscillatory initial data: we are able to deal with larger oscillations. This is joint work with Ivar Ekeland (JEMS 23 no. 10, 2021).

  • Birgit Schorkhuber - Nontrivial self-similar blowup for the supercritical quadratic wave equation

    This talk will be concerned with the quadratic wave equation in high space dimensions where the model is energy supercritical. We present a new nontrivial self-similar solution, which is completely explicit in all supercritical dimensions. This solution blows up at a single point and continues naturally past the blowup time. We prove that this solution is co-dimension one stable, locally in a backward light cone as well as in a larger spacetime region that extends arbitrarily close to the future light cone of the singularity. The talk is based on recent joint works with Irfan Glogić, Elek Csobo, Michael McNulty, Roland Donninger and Po-Ning Chen.

  • Gigliola Staffilani - On the wave turbulence theory for a stochastic KdV type equation

    This talk is a summary of a recent work completed with Binh Tran. Starting from the stochastic Zakharov-Kuznetsov (ZK) equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a derivation of the 3-wave kinetic equation. We show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension is d>1, the smallness of the nonlinearity is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. Unlike the cubic nonlinear Schr ̈odinger equation, for which such a general result is commonly expected without the noise, the kinetic description of the deterministic lattice ZK equation is unlikely to happen. One of the key reasons is that the dispersion relation of the lattice ZK equation leads to a singular manifold, on which not only 3-wave interactions but also all m-wave interactions are allowed to happen. To the best of our knowledge, the work provides the first rigorous derivation of nonlinear 3-wave kinetic equations. Also this is the first derivation for wave kinetic equations in the lattice setting and out-of-equilibrium.

  • Benjamin Texier - Weak and strong defects of hyperbolicity

    I'll talk about the lack of well-posedness or stability induced by defects of hyperbolicity in systems of quasi-linear partial differential equations. Even weak defects of hyperbolicity, which are bounded in the frequency domain, or occur on the boundary of the region of hyperbolicity in the frequency domain, generate strong instabilities. I'll give examples on physical systems from plasma physics, and examples of formulations of these defects in general first-order systems.
    This is based on joint work with Federico Cacciafesta, Eric Dumas, Nicolas Lerner, Lu Yong, Toan Nguyen and Marta Strani.

  • Nicola Visciglia - Global existence for NLS with multiplicative white noise on T^2

    We discuss a series of joint works with N. Tzvetkov about the global existence of solutions to NLS with multiplicative white noise and with a generic nonlinearity on T^2. Time permitting we also discuss some recent extensions in the euclidean setting R^2 (in collaboration with A. Debbusche and N. Tzvetkov).

  • Luis Vega - New Conservation Laws and Energy Cascade for 1d Cubic NLS

    I’ll present some recent results concerning the IVP of 1d cubic NLS at the critical level of regularity. I’ll also exhibit a cascade of energy for the 1D Schrödinger map which is related to NLS through the so called Hasimoto transformation. For higher regularity these two equations are completely integrable systems and therefore no cascade of energy is possible.

  • Kevin Zumbrun - Large-amplitude modulation of periodic traveling waves (joint with G. Metivier)

    We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit. This result is new for either large amplitudes or multi-d. Applications to pattern formation are many, and well known (Howard-Kopell, Doelman-Sandstede-Scheel-Schneider, ...). The method suggests possibilities also to treat more complicated phenomena in shallow-water and thin-film flow.