Abstract: I will present a result, in collaboration with Benoît Grébert and Tristan Robert, concerning the existence of infinite-dimensional invariant tori for nonlinear Schrödinger equations on the circle (without external parameters). In particular, we prove the existence of almost-periodic solutions that are not quasi-periodic. Our proof relies in particular on KAM theory and dispersive regularizing effects, which I will describe.
Abstract: The stability of traveling periodic Stokes waves—the first global-in-time solutions ever discovered for nonlinear quasi-linear dispersive PDEs—is a central, long-standing question in fluid mechanics. Do these fundamental solutions remain stable or do they break down? Historically, in 1967, Benjamin and Feir proposed a famous heuristic mechanism suggesting instability under longitudinal long-wave perturbations, later complemented by McLean's numerical identification of additional instability mechanisms. This talk will present a mathematically complete description of the Floquet spectral bands for the linearized water wave operator at small-amplitude Stokes waves. We achieve this by exploiting the Hamiltoinian and reversible strucure inherent to this system and rigorously combining spectral theory, perturbation theory, and dynamical systems techniques.
Abstract: In this talk, I will present recent results on energy cascades and structure formation in Hamiltonian systems. I will introduce two families of solvable models that explicitly demonstrate the dynamical development of energy cascades and the emergence of both large- and small-scale structures. Some solutions describe condensate formation driven by highly coherent dynamics, while all cascade solutions exhibit the finite-time formation of power-law energy spectra, leading to the blow-up of Sobolev norms.
Abstract: This talk presents a probabilistic approach to the cubic nonlinear Schrödinger equation (NLS) on the two-dimensional sphere, focusing on the collective behavior of random initial data distributed according to the Gibbs measure.
We begin with a brief overview of the Cauchy problem for NLS on compact surfaces. Then, we introduce the Gibbs measure problem and Bourgain’s resolution scheme in the case of the flat torus.
On the sphere, we show that strong instabilities arise due to the concentration of spherical harmonics around great circles, preventing a direct extension of Bourgain’s method.
To address these instabilities, we develop a non-perturbative resolution scheme, building on recent works by Bringmann, Deng, Nahmod and Yue.
This talk is based on a series of joint works with Nicolas Burq, Chenmin Sun, and Nikolay Tzvetkov.
Abstract: We will begin with some reminders on the phenomenology of three-dimensional fluid turbulence and the observational aspects of the statistical behavior of solutions of the Navier-Stokes equations, forced by a smooth term in space. To account for the cascade phenomenon, which asymptotically leads to a velocity field that is continuous yet highly irregular (of Hölderian type), we propose a linear model dynamics capable of generating such rough fields from entirely smooth ingredients over infinite time. This will involve a theoretical and numerical study of a transport phenomenon, not in physical space but in Fourier space, allowing energy to be transferred (or cascaded) across scales. A scheme based on finite spectral volumes will then provide a coherent numerical representation of such dynamics. This is joint work with G. Apolinario, G. Beck, C.-E. Bréhier, I. Gallagher, R. Grande, J.-C. Mourrat and W. Ruffenach.
Abstract: I shall discuss the existence of infinite-dimensional invariant tori in a mechanical system made of infinitely many rotators weakly interacting with each other. I shall concentrate on interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy.
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 joint work with Annalaura Stingo (Ecole Polythecnique) and Arthur Touati (Bordeaux).
Abstract: We consider the quantum hydrodynamics system in a 2-dimensional periodic setting and discuss the existence of wave solutions which display energy transfer to high Fourier modes. These solutions are uniformly far from vacuum, suggesting that weak turbulence phenomena in quantum hydrodynamics are not necessarily related to the occurrence of vortex structures.
We detect energy cascades by measuring the inflation of high order Sobolev norms and show that the time at which these norms exhibit a large growth is uniform when approaching the semiclassical limit to the compressible Euler equations.
Abstract: In this talk I will present a recent result concerning the long time dynamics for the 3D pure gravity water waves equations in finite depth. This is a quasi-linear dispersive PDE in dimension 2 (written in suitable coordinates) with dispersion relation of order 1/2.
Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics.
I shall discuss a result concerning the existence of global in time solutions (that are not stationary in any reference frame) that are approximately finite sums (arbitrarily big) of traveling wave solutions oscillating with frequencies that are rationally independent (they satisfies suitable non-resonance conditions).
This is a very hard problem due to the fact that one deals with a dispersive quasi-linear PDE in higher dimension. The main problems are:
The geometry of the resonances is very complicated
The control of the derivatives in the nonlinearity in the higher dimensional framework.
I shall describe a novel strategy that allows to perform the spectral analysis of the linearized equations (at any approximate traveling wave solutions) which is suited for higher dimensional dispersive PDEs with sublinear dispersion and the usage of conservation laws such as the momentum in order to get some cancellation of resonant terms.
Our result is the first KAM result for an autonomous quasi-linear PDE in dimension greater than one and it solves a long standing open problem in Fluid Mechanics and in the community of KAM for PDEs.
This is a joint work with R. Feola and S. Terracina.
Abstract: In this talk, I will present recent experimental findings aimed at testing the predictions of Wave Turbulence Theory for surface gravity waves. In particular, I will provide evidence that resonant and quasi-resonant interactions play a key role in the transfer of energy across different wave scales in the ocean. These results have been made possible through the use of stereoscopic measurement techniques, which enable the reconstruction of the sea surface elevation in both space and time.
Abstract: The FPUT problem is framed in the theory of quasi-integrable, many degrees of freedom Hamiltonian systems, with the aim to show that the unobserved trend to the equilibrium predicted by statistical mechanics is just due to a separation of time-scales. The main focus will be on the nonlinear wave equations providing the local normal form dynamics of the system on the short term and allowing to predict some numerically observed phenomena with significant precision.
Abstract: In the study of close to integrable Hamiltonian PDEs, a fundamental question is to understand the behaviour of “typical” solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are in fact typical in the integrable case. In this talk I shall consider a family of NLS equations parametrized by a smooth convolution potential and prove that for “most” choices of the parameter there is a full measure set of Gevrey initial data that give rise to almost-periodic solutions whose hulls are invariant tori. As a consequence the elliptic fixed point at the origin turns out to be statistically stable in the sense of Lyapunov. This is a joint work with L. Biasco, G. Gentile and L. Corsi.
Abstract: Fluid dynamics presents significant computational challenges due to its highly nonlinear and nonlocal nature. Many fluid systems can be reformulated as quasi-linear, nonlocal contour dynamics equations (CDEs) that describe the evolution of sharp interfaces. This talk outlines new and flexible paralinearization techniques for such CDEs and their application in proving long-time existence results for several quasi-linear models.
Abstract: In this talk we construct smooth, non-radial solutions of the defocusing nonlinear Schrödinger equation that develop an imploding finite time singularity, both in the periodic setting and the full space. The result is obtained by transforming the NLS into a compressible Euler type equation via the Madelung transformation and use imploding solutions for them. This is joint work with Gonzalo Cao, Javi Gomez Serrano and Jia Shi.
Abstract: We present here a mean-field theory for the Fermi-Pasta-Ulam-Tsingou model with quartic interaction: a non-perturbative approach is proposed which is exact in the thermodynamic limit and is capable to capture the salient features of the model at all energies scales. We show that there is a mean-field Hamiltonian H_mf which, considered as a random variable with distribution induced by the Boltzmann one, is close to the original Hamiltonian in the limit N >> 1. We find that this H_mf generates, in the N>>1 limit, the dynamics of N independend renormalized Fourier modes with frequency $\tilde{\omega}^2_k = \omega_k^2(1+\alpha_\beta)$, where $\omega_k^2$ is the original frequency of Fourier modes and $\alpha_\beta$ is a function of the temperature which can determined exactly. Analytical predictions drawn from an effective Langevin dynamics of independent renormalized Fourier modes are successfully compared against data of the original Hamitonian dynamics. The whole procedure holds, despite the presence of chaos, in the limit $N\gg 1$, and it is fully non-perturbative.
Work in collaboration with M. Baldovin, G. Gradenigo and A. Ponno
Abstract: Vortex filaments that evolve according the binormal flow are expected to exhibit turbulent properties. Aiming to quantify this, I will discuss the multifractal properties of the family of functions
R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, x_0 \in [0,1],
that approximate the trajectories of regular polygonal vortex filaments. These functions are a generalization of the classical Riemann's non-differentiable function, which we recover when $x_0 = 0$. I will highlight how the analysis seems to critically depend on $x_0$, and I will discuss the important role played by Gauss sums, a restricted version of Diophantine approximation, the Duffin-Schaeffer theorem, and the mass transference principle.
This talk is based on the article https://doi.org/10.1007/s00208-024-02971-0 in collaboration with Valeria Banica (Sorbonne Université), Andrea R. Nahmod (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU).
Abstract: In this talk, I will present a recent result on the existence of almost-periodic solutions for the incompressible Euler equations on the d-dimensional periodic torus, with d=3 or d even. The construction of such solutions extends previous results for quasi-periodic solutions by Crouseille and Faou (in dimension 2) and Enciso, Peralta Salas and Torres de Lizaur (in dimensions 3 or even). It turns out, however, that in the almost-periodic setting nontrivial questions on the regularity of the embedding of the infinite dimensional torus and of the velocity field arise, which were not an issue in in the previous works.
This is a joint work with Riccardo Montalto.
Abstract: We study the transfer of energy to high frequencies in a quasi-linear Schrödinger equation with a sublinear dispersion relation on the one-dimensional torus, a toy model for gravity water waves. We construct initial data that exhibit finite-time Sobolev norm growth: starting with arbitrarily small norms in high-regularity spaces, these norms become arbitrarily large at later times.
Our analysis identifies an instability mechanism driving this energy cascade. Using paradifferential normal forms, we derive an effective equation governed by a transport operator with non-constant coefficients. A positive commutator method inspired by Mourre’s theory reveals how this operator induces the instability.
We believe that our work provides a foundational step toward investigating energy cascades in more complex fluid models, including gravity water waves.
Abstract: We consider a two-dimensional incompressible inviscid fluid with variable density, under the action of gravity. We assume that the equilibrium density profile is stable, and we consider the so-called Boussinesq approximation, which neglects density variations in terms which do not involve gravity; this system has been widely used in physical literature to describe internal gravity waves. We prove a modulational instability result for such system, namely that the linearization at a small-amplitude travelling wave admits at least an eigenvalue with positive real part. This can be regarded as a rigorous proof of the phenomenon of triadic resonant instability (TRI) for inviscid fluids, in which one observes a transfer of energy between a primary initially excited wave and two secondary waves with different frequencies.
This is a joint work with A. Maspero (SISSA) and R. Bianchini (CNR).
Abstract: The goal of this talk is to discuss the existence of large amplitude traveling waves of the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with a traveling wave external force. More precisely, we assume that the force is a smooth bi-periodic traveling wave propagating in the direction $\omega=(\omega_{1}, \omega_{2})\in\mathbb{R}^{2}$, with large amplitude of order $O(\lambda^{1^{+}})$ and with large velocity speed $\lambda\omega$.
Then, for most values of $\omega$ and for $\lambda\gg1$ large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude. Due to the presence of small divisors, the proof is based on a nonlinear Nash-Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator and hence the problem is not perturbative. The inversion of the linearized operator is then performed by using tools from micro-local analysis and normal form together with a sharp analysis of high and low frequency regimes with respect to the large parameter $\lambda$.
This works offers the first existence result for large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimensions. This research extends KAM techniques to a complex scenario in fluid mechanics.This is a joint work with G. Ciampa and R. Montalto.