This workshop is centered around wave propagation in complex environnements, and how its dynamics may be related to both the medium and nonlinear effets. It brings together people working at the crossroads between dynamical systems, nonlinear PDEs, control theory, numerical analysis and microlocal analysis.
Talks
Jacopo Bellazzini
Compact embeddings for fractional super and sub harmonic functions with radial symmetry
Abstract : We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite L2 norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions. Joint work with Vladimir Georgiev.
Rémi Carles
Scattering for NLS : rigidity properties and numerical simulations via lens transform
Abstract : We recall some more or less well-known facts on the scattering operator associated to the defocusing nonlinear Schrodinger equation, and establish some apparently new rigidity properties. We use these properties to secure numerical simulations relying on the lens transform. The results suggest some conjectures regarding the dynamical behavior of the scattering operator.
Andreia Chapouto
Deep- and shallow-water limits of statistical equilibria for the intermediate long wave equation
Abstract : The intermediate long wave equation (ILW) models the internal wave propagation of the interface in a stratified fluid of finite depth, providing a natural connection between the deep-water regime (= the BO regime) and the shallow-water regime (= the KdV regime).
Exploiting the complete integrability of ILW, I will discuss the statistical convergence of ILW to both BO and KdV, namely the convergence of the higher order conservation laws for ILW and their associated invariant measures.
In particular, as KdV possesses only half as many conservation laws as ILW and BO, we observe a novel 2-to-1 collapse of ILW conservation laws to those of KdV, which yields alternative modes of convergence for the associated measures in the shallow-water regime.
This talk is based on joint work with Guopeng Li and Tadahiro Oh.
Charles Collot
The Gross-Pitaevskii equation linearized around a vortex
Abstract : The Gross-Pitaevskii equation admits in two dimensions stationary solutions, among them the Ginzburg-Landau vortex of degree one. Its orbital stability was proved by Gravejat-Pacherie-Smets. We obtain its linear asymptotic stability, via a detailed spectral study of the linearized operator, through the description of its spectrum, of its generalized eigenfunctions, and of the properties of the associated distorted Fourier transform. This is joint work with P. Germain (Imperial College) and E. Pacherie (CY Cergy Paris Université).
Anne-Sophie de Suzzoni
Trivial resonances for a system of Klein-Gordon equations and statistical applications
Abstract : In the derivation of the wave kinetic equation coming from the Schrödinger equation, a key feature is the invariance of the Schrödinger equation under the action of U(1). This allows quasi-resonances of the equation to drive the effective dynamics of the statistical evolution of solutions to the Schrödinger equation. In this talk, I will give an example of an equation that does not have the same invariance as the Schrödinger equation, and I will show that in this example, exact resonances (always) take precedence over quasi-resonances, so that the effective dynamics of the statistical evolution of the solutions are not kinetic. However, these dynamics are not linear (let alone trivial). I will present the problem and the ideas involved in deriving the effective dynamics and some elements of proof: in particular, I will describe the representation of solutions of the initial equation in diagrammatic form. This talk is based on a work in progress with Annalaura Stingo (X) and Arthur Touati (IHES/Bordeaux).
Luca Fanelli
Intertwining operators beyond the Stark Effect
Abstract: We establish a general framework in which some suitable intertwining operators can be defined for non constant spherical perturbations in space dimensions 2 and higher, according to the well known Stark Effect from Quantum Mechanics. In addition, we investigate the mapping properties between Lp-spaces of these operators. In 2D, we prove a complete result, for the Schrödinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and Lp-bounds of Bochner–Riesz means.
The work is in collaboration with X. Su, Y. Wang, J. Zhang, and J. Zheng.
Erwan Faou
On the energy spectrum of wave equations with forcing
Abstract : We consider a problem widely studied by theoretical and experimental physicists. When a wave-like device (quantum solitons, water waves, etc…) is forced by a smooth term concentrated in low frequencies, a frequency cascade is created at a rate following power laws with esoteric exponents. In this talk, we propose a new explanation of these exponents based on singularity formation analysis. We give some rigorous examples of cascade and study their stabilities. This is a joint work with R. Carles (CNRS, Rennes).
Mahir Hadzic
On quantitative gravitational relaxation
Abstract: We obtain quantitative decay rates for the linearised gravitational potential around compactly supported steady states of the Vlasov-Poisson system featuring a point mass potential at the origin. Such steady states feature stably trapped particles which present a severe obstacle to any kind of dispersion. The problem is further complicated by the presence of an infinite-dimensional kernel. To handle these issues we combine tools from dynamical systems, Hamiltonian geometry, and scattering theory. Our theorem can be viewed as a first quantitative proof of (linear) gravitational Landau damping. Joint work with Matthew Schrecker.
Ryan Kim
On the regularity of the self-similar profiles
Abstract: We prove smoothness of self-similar profiles in two different models. Generically, these profiles have limited regularity at the singular point of the ODE. The aim is to prove existence of a sequence of values of the parameter so that the resulting solution is smooth.
Firstly, we prove this for the self-similar equation for the compressible Euler model. The vanishing eye structure (two singular points colliding in some limit) of the phase portrait is used in a fundamental way in the proof and it is responsible for the leading order behaviour of the Taylor coefficients at these points.
Secondly, we consider a different model where there is no vanishing eye. In this talk, we outline the main idea of proof and difference in the methods used in the two problems.
David Lafontaine
Scattering for defocusing cubic NLS under locally damped strong trapping
Abstract : We will be interested in the scattering problem for the cubic 3D nonlinear defocusing Schrödinger equation with variable coefficients. Previous scattering results for such problems address only the cases with constant coefficients or assume strong non-trapping conditions. In contrast, we consider the most general setting, where strong trapping, such as stable closed geodesics, may occur, but we introduce a compactly supported damping term localized in the trapping region, to explore how damping can mitigate the effects of trapping.
In addition to the challenges posed by the trapped trajectories, notably the loss of smoothing effect and of scale-invariant Strichartz estimates, difficulties arise from the damping itself, particularly since the energy is not, a priori, bounded. For H1+epsilon initial data – chosen because the local-in-time theory is a priori no better than for 3D unbounded manifolds, where local well-posedness of strong H1 solutions is unavailable – we establish global existence and scattering in H1/2 in positive times, the loss of regularity in scattering being related to the loss of smoothing due to trapping.
Joint work with Boris Shakarov (Toulouse).
Camille Laurent
Propagation of global analyticity and unique continuation for semilinear wave equations
Abstract : In this talk, I will first present some known results of unique continuation for wave-like equations. I will explain the difficulties of obtaining global results under natural geometrical assumptions. Then, I will present a recent result, in collaboration with Cristobal Loyola, where we prove unique continuation for semilinear wave equations under the geometric control assumption. A crucial step is the global propagation of analyticity in time from open sets verifying the geometric control condition. The proof uses control methods associated with Hale-Raugel ideas concerning attractor regularity.
Eliot Pacherie
Orbital stability of the vortex pair for the Gross-Pitaevskii equation
Abstract : The Gross-Pitaevskii equation is a nonlinear Schrödinger equation with a nonzero condition at infinity. The equation admits stationnary solution called vortices. When two such vortices are present, they move together at a constant speed. In this talk, we will show the orbital stability in a metric space of this travelling wave solution. We will explain how to adapt the classical scheme of proof of orbital stability in such a space, and why the proof fails in more classical settings. This is a joint work with Philippe Gravejat and Frederic Valet.
Frédéric Rousset
Semiclassical limit of NLS and Hartree type equations for mixed states
Abstract : We study the semiclassical limit of NLS and Hartree type equations for infinitely many particles in the case that the Wigner transform of the initial datum is smooth. The formal limit is a singular kinetic equation of Vlasov type where the force field is given by the gradient of the density.
We will recall previous results on the well-posedness of this type of kinetic equations and explain how to get uniform estimates suitable to justify the semiclassical limit.
Joint work with D. Han-Kwan (Nantes) and T. Chaub (Orsay).
Diego Sanchez Sanz
Dispersion for the wave equation with Robin boundary conditions on a 2d convex model domain
Abstract : In the last years, several developments have been done on the study of Strichartz estimates for the wave equation in convex domains.
However, this developments have always been done for Dirichlet boundary conditions. It seems natural to ask if the same techniques can be used to other boundary conditions. In the case of Neumann, the answer is (relatively) straightforward. However, in the Robin case, some adaptations must be made.
We introduce here how to aproach the Robin problem in the ilustrative example of the Friedlander model, focusing in the differences with the Dirichlet problem as well as how to solve them.
Katharina Schratz
Resonances as a computational tool
Abstract : A large toolbox of numerical schemes for dispersive equations has been established, such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). These classical schemes are based on linearised time dynamics and in many situations allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.
Luis Vega
Fluctuations of delta-moments for Schrödinger and Helmholtz Equations
Abstract : I will present recent work done with J. Canto, S. Kumar, F. Ponce-Vanegas, L. Roncal and N. Schiavone. In the first part of the talk we study the process of dispersion of low-regularity solutions to the free Schrödinger equation using fractional weights. We give another proof of the uncertainty principle for fractional weights and use it to get a lower bound for the concentration of mass. We consider also the evolution when the initial datum is the Dirac comb in the real line. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a Lévy process. Furthermore, the evolution exhibits intermittency and multifractality. In the second part we will show how these results can be extended to solutions of the Helmholtz equation.
Zoé Wyatt
A new phase transition in cosmological fluid dynamics
Abstract : On flat geometries, the Euler equations (both relativistic and not) are known to admit unstable homogeneous solutions with finite-time shock formation. In cosmological settings, the spatial geometry expands at a particular rate a(t) with a'(t)>0. This leads to a competition between dissipation (from the expansion rate a(t)) and shock formation (from nonlinear advection terms). I will present some recent joint work in this direction, and a novel phase transition which arises in decelerated cosmological settings.