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Organizing Committee
Paris-Sud University, Orsay
NYU - Courant Institute
Duke University
University of Massachusetts Amherst
Abstract
The large-time behavior of (generic) solutions of nonlinear dispersive equations set on bounded domains is almost completely open as far as rigorous analysis goes, and fairly mysterious, even from a less rigorous viewpoint. Under the assumption of weak nonlinearity, physicists and applied mathematicians have devised a theory to approach this question, known as weak turbulence, a branch of statistical physics. Weak turbulence theory predicts that the equation will enter a chaotic regime, where the exchange of energy in phase space is governed by the so-called kinetic wave equation. Justifying the derivation of the kinetic wave equation is a fascinating mathematical task, for which some results are already known, but whose solution will likely require input from nonlinear PDEs, but also probability theory. Intimately related questions are the question of Sobolev growth (how much can or does, the Sobolev norm of a nonlinear dispersive equation grow over time), as well as the analysis of nonlinear dispersive equations with random data. These questions have already generated a lot of interest, but much remains to be discovered: our understanding is extremely lacunar.
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Workshop Schedule
Monday, October 18, 2021
8:45 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:50 AM EDT
Microlocal analysis of singular measures
11th Floor Lecture Hall
Virtual Speaker
Nicolas Burq, University Paris-Sud
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:20 AM EDT
Full description of Benjamin-Feir instability of Stokes waves in deep water
11th Floor Lecture Hall
Virtual Speaker
Alberto Maspero, Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Abstract
Small-amplitude, traveling, space periodic solutions -- called Stokes waves -- of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave 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.
11:30 AM - 1:00 PM EDT
Lunch/Free Time
1:00 - 1:50 PM EDT
Breakdown of small amplitude breathers for the nonlinear Klein-Gordon equation
11th Floor Lecture Hall
Virtual Speaker
Marcel Guardia, Universitat Politècnica de Catalunya
Abstract
Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arise from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study the nonlinear Klein-Gordon equation and show that small amplitude breathers cannot exist (under certain conditions). We also construct generalized breathers, these are solutions which are periodic in time and in space are localized up to exponentially small (with respect to the amplitude) tails. This is a joint work with O. Gomide, T. Seara and C. Zeng.
2:00 - 2:50 PM EDT
Fluctuations of \delta-moments for the free Schrödinger Equation
11th Floor Lecture Hall
Luis Vega, Basque Center for Applied Mathematics (BCAM)
Abstract
I will present recent work done with S. Kumar and F.Ponce-Vanegas.
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 R. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a Lévy process. Furthermore, the evolution exhibits multifractality.
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 R. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a Lévy process. Furthermore, the evolution exhibits multifractality.
3:00 - 4:30 PM EDT
Welcome Reception
Reception - Hemenway's Patio
Tuesday, October 19, 2021
9:00 - 9:50 AM EDT
Mathematical wave turbulence and propagation of chaos (I)
11th Floor Lecture Hall
Yu Deng, University of Southern California
Abstract
The theory of wave turbulence can be traced back to the 1920s and has played significant roles in many different areas of physics. However, for a long time the mathematical foundation of the theory has not been established. The central topics here are the wave kinetic equation, which describes the thermodynamic limit of interacting wave systems, and the propagation of chaos, which is a fundamental physical assumption in this field that lacks mathematical justification. In this first talk, I will present recent results with Zaher Hani (University of Michigan), where we provide the first rigorous derivation of the wave kinetic equation, and also justify the propagation of chaos assumption in the same setting. In part (II), we will discuss some important ideas in the proof.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:20 AM EDT
Mathematical wave turbulence and propagation of chaos (II)
11th Floor Lecture Hall
Zaher Hani, University of Michigan
Abstract
The theory of wave turbulence can be traced back to the 1920s and has played significant roles in many different areas of physics. However, for a long time the mathematical foundation of the theory has not been established. The central topics here are the wave kinetic equation, which describes the thermodynamic limit of interacting wave systems, and the propagation of chaos, which is a fundamental physical assumption in this field that lacks mathematical justification. This talk is a continuation of that of Yu Deng (University of Southern California) who will present our recent joint results that provide the first rigorous derivation of the wave kinetic equation, and also justify the propagation of chaos assumption in the same setting. In this second part, we will discuss some important ideas in the proof.
11:30 AM - 1:00 PM EDT
Lunch/Free Time
1:00 - 1:50 PM EDT
Energy transfer for solutions to the nonlinear Schrodinger equation on irrational tori.
11th Floor Lecture Hall
Gigliola Staffilani, Massachusetts Institute of Technology
Abstract
We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schr\"odinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.
2:00 - 2:50 PM EDT
Determinants, commuting flows, and recent progress on completely integrable systems
11th Floor Lecture Hall
Virtual Speaker
Monica Visan, University of California, Los Angeles
Abstract
We will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE. These include a priori bounds, the orbital stability of multisolitons, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.
3:10 - 4:00 PM EDT
Lightning Talks
11th Floor Lecture Hall
4:00 - 4:30 PM EDT
Coffee Break
11th Floor Collaborative Space
Wednesday, October 20, 2021
9:00 - 9:50 AM EDT
On the derivation of the Kinetic Wave Equation in the inhomogeneous setting
11th Floor Lecture Hall
Virtual Speaker
Charles Collot, Cergy-Paris Université
Abstract
The kinetic wave equation arises in weak wave turbulence theory. In this talk we are interested in its derivation as an effective equation from dispersive waves modelled with quadratic nonlinear Schrodinger equations. We focus on the space-inhomogeneous case, which had not been treated earlier. More precisely, we will consider such a dispersive equations in a weakly nonlinear regime, and for highly oscillatory random Gaussian fields with localised enveloppes as initial data. A conjecture in statistical physics is that there exists a kinetic time scale on which, statistically, the Wigner transform of the solution (a space dependent local Fourier energy spectrum) evolve according to the kinetic wave equation.
I will present a joint work with Ioakeim Ampatzoglou and Pierre Germain (Courant Institute) in which we approach the problem of the validity of this kinetic wave equation through the convergence and stability of the corresponding Dyson series. We are able to identify certain nonlinearities, dispersion relations, and regimes, and for which the convergence indeed holds almost up to the kinetic time (arbitrarily small polynomial loss).
I will present a joint work with Ioakeim Ampatzoglou and Pierre Germain (Courant Institute) in which we approach the problem of the validity of this kinetic wave equation through the convergence and stability of the corresponding Dyson series. We are able to identify certain nonlinearities, dispersion relations, and regimes, and for which the convergence indeed holds almost up to the kinetic time (arbitrarily small polynomial loss).
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:20 AM EDT
3 Problems in Wave Turbulence
11th Floor Lecture Hall
Jalal Shatah, New York University
11:30 - 11:40 AM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
11:40 AM - 1:00 PM EDT
Lunch/Free Time
1:00 - 1:50 PM EDT
Constructing global solutions for energy supercritical NLS equations
11th Floor Lecture Hall
Virtual Speaker
Mouhamadou Sy, Imperial College London
Abstract
The last decades were very fruitful for the realm of dispersive PDEs. Besides several new deterministic developments in the study of the initial value problem and the behavior of solutions, probabilistic methods were introduced and made important progresses, particularly on bounded domain settings. Invariant measure are of considerable interest in these questions. However, in the context of energy supercritical equations, both the well-known Gibbs measures based strategy and the standard fluctuation-dissipation approach come across serious limitations. In this talk, we will present a new approach that combines the aforementioned ones to construct invariant measures, almost sure GWP, and strong controls on the time evolution of the solutions for the periodic NLS, with arbitrarily large power of nonlinearity and in any dimension. We will discuss the application to other contexts including non-dispersive PDEs.
2:00 - 2:30 PM EDT
Coffee Break
11th Floor Collaborative Space
2:30 - 3:20 PM EDT
Invariant Gibbs measures for NLS and Hartree equations
11th Floor Lecture Hall
Haitian Yue, University of Southern California
Abstract
In this talk, I'll present our results about invariant Gibbs measures for the periodic nonlinear Schrödinger equation (NLS) in 2D, for any (defocusing and renormalized) odd power nonlinearity and for the periodic Hartree equation in 3D. The results are achieved by introducing a new method (we call the random averaging operators method) which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Andrea Nahmod (UMass Amherst).
Thursday, October 21, 2021
9:00 - 9:50 AM EDT
Singularities in the weak turbulence regime
11th Floor Lecture Hall
Virtual Speaker
Anne-Sophie de Suzzoni, Ecole Polytechnique
Abstract
In this talk, we discuss the different regimes for the derivation of kinetic equations from the theory of weak turbulence for the quintic Schrödinger equation. In particular, we see that there exists a specific regime such that the correlations of the Fourier coefficients of the solution of the Schrödinger equation converge (in this regime) to a function that has an inifinite number of discontinuities.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:20 AM EDT
A New model for the stochasticly perturbed 2d navier-stokes equations
11th Floor Lecture Hall
Jonathan Mattingly, Duke University
Abstract
I will introduce a new model of the stochastically forced navier-stokes equation. The model will be targeted at studying the Equations forced by a large scale forcing. I give a number of properties of the model.
11:30 AM - 1:00 PM EDT
Lunch/Free Time
1:00 - 1:50 PM EDT
Global well-posedness for the fractional NLS on the unit disk
11th Floor Lecture Hall
Xueying Yu, University of Washington
Abstract
In this talk, we discuss the cubic nonlinear Schr\"odinger equation with the fractional Laplacian on the unit disk. We show the global well-posedness for certain radial initial data below the energy space and establish a polynomial bound of the global solution. The result is proved by extending the I-method in the fractional nonlinear Schr\"odinger equation setting.
2:00 - 2:30 PM EDT
Coffee Break
11th Floor Collaborative Space
2:30 - 3:20 PM EDT
The wave maps equation and Brownian paths
11th Floor Lecture Hall
Bjoern Bringmann, Institute for Advanced Study
Abstract
We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Riemannian manifold $\mathcal{M}$. Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold $\mathcal{M}$ as initial data. Our main theorem is the probabilistic local well-posedness of the associated initial value problem. The analysis in this setting involves analytic, geometric, and probabilistic aspects. This is joint work with J. Lührmann and G. Staffilani.
Friday, October 22, 2021
9:00 - 9:50 AM EDT
Positive Lyapunov exponents for the Galerkin-Navier-Stokes equations with stochastic forcing
11th Floor Lecture Hall
Jacob Bedrossian, University of Maryland
Abstract
In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:20 AM EDT
A tale of two generalizations of Boltzmann equation
11th Floor Lecture Hall
Nataša Pavlović, University of Texas at Austin
Abstract
In the first part of the talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. This part of the talk is based on the joint work with Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. In the second part of the talk we will discuss a rigorous derivation of a Boltzmann equation for mixtures of gases, which is a recent joint work with Ampatzoglou and Miller. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases.
11:30 AM - 12:20 PM EDT
Some Recent Results On Wave Turbulence: Derivation, Analysis, Numerics and Physical Application
11th Floor Lecture Hall
Minh-Binh Tran, Southern Methodist University
Abstract
Wave turbulence describes the dynamics of both classical and non-classical nonlinear waves out of thermal equilibrium. In this talk, we will discuss some of our recent results on some aspects of wave turbulence, concerning the derivation and analysis of wave kinetic equations, some numerical algorithms and physical applications in Bose-Einstein Condensates.
