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Organizing Committee
Princeton University
École Polytechnique
Research Institute for Mathematical Sciences, Kyoto University
University of California, Los Angeles
Abstract
Recent progress in the analysis of dispersive PDE's has revealed various aspects of long-time dynamics or behavior of solutions, from the basic three types (scattering, blow-up, and solitons) to more complicated combinations, transitions, and oscillations among them, and so on. The goal of this workshop is for the participants to draw integrated landscapes of those diverse phenomena, aiming towards more a complete description, classification, and prediction of global dynamics, as well as new phenomena and methods.
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Workshop Schedule
Monday, December 06, 2021
8:55 - 9:00 AM EST
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 10:00 AM EST
Asymptotic stability of the Sine-Gordon kink under odd perturbations via super-symmetry
11th Floor Lecture Hall
Wilhelm Schlag, Yale University
Abstract
We will describe the recent asymptotic analysis with Jonas Luehrmann of the Sine-Gordon evolution of odd data near the kink. We do not rely on the complete integrability of the problem in a direct way, in particular we do not use the inverse scattering transform.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Time-dependent Bogoliubov-de-Gennes and Ginzburg-Landau equations
11th Floor Lecture Hall
Virtual Speaker
Rupert Frank, LMU Munich
Abstract
We study the time-dependent Bogoliubov--de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg--Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time.
11:30 AM - 12:15 PM EST
On the well-posedness of the derivative nonlinear Schr\"odinger equation
11th Floor Lecture Hall
Maria Ntekoume, Rice University
Abstract
We consider the derivative nonlinear Schr\"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. However, not much is known regarding the well-posendess of the equation below $H^{\frac 12}$. In this talk, we prove that this problem is globally well-posed for initial data in the Sobolev spaces $H^s$ for $\frac 1 6\leq s<\frac 12$. The key ingredient in our argument is proving that ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. This is joint work with Rowan Killip and Monica Visan.
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:30 PM EST
Quantitative derivation and scattering of the 3D cubic NLS in the energy space
11th Floor Lecture Hall
Justin Holmer, Brown University
Abstract
We consider the derivation of the {defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under $H^{1}$ regularity. The $H^{1}$ convergence rate estimate we obtain is almost optimal for $H^{1}$ datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state. This is joint work with Xuwen Chen (University of Rochester).
3:30 - 4:30 PM EST
Local smoothing estimate for the cone in R^3
11th Floor Lecture Hall
Virtual Speaker
Hong Wang, Institute for Advanced Study (IAS)
Abstract
If u is a solution to the wave equation on R^n, a local smoothing inequality bounds $\|u\|_{L^p(\mathbb{R}^n\times [1,2])}$ in terms of the Sobolev norms of the initial data. We prove Sogge's local smoothing conjecture in 2+1 dimensions.
In the proof, we introduced an approximation of the $L^4$--norm that works better for induction. Another key ingredient is an incidence estimate for points and tubes.
This is joint work with Larry Guth and Ruixiang Zhang.
In the proof, we introduced an approximation of the $L^4$--norm that works better for induction. Another key ingredient is an incidence estimate for points and tubes.
This is joint work with Larry Guth and Ruixiang Zhang.
4:30 - 6:00 PM EST
Welcome Reception
Reception - 11th Floor Collaborative Space
Tuesday, December 07, 2021
9:00 - 9:45 AM EST
High-Order Rogue Waves and Solitons, and Solutions Interpolating Between Them
11th Floor Lecture Hall
Virtual Speaker
Peter Miller, University of Michigan
Abstract
A family of exact solutions to the focusing nonlinear Schrödinger equation is presented that contains fundamental rogue waves and multiple-pole solitons of all orders. The family is indexed with a continuous parameter representing the "order" that allows one to continuously tune between rogue waves and solitons of different integer orders. In this scheme, solitons and rogue waves of increasing integer orders alternate as the continuous order parameter increases. For example, the Peregrine solution can be viewed as a soliton of order three-halves. We show that solutions in this family exhibit certain universal features in the limit of high (continuous) order. This is joint work with Deniz Bilman (Cincinnati).
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Landau law and stability of 3D shocks
11th Floor Lecture Hall
Virtual Speaker
Igor Rodnianski, Princeton University
Abstract
We will discuss stability and long term behavior of 3 dimensional compressible irrotational shocks arising in the compressible Euler equations.
In 1945, Landau argued that spherically symmetric solutions which form weak shocks will settle to a profile with 2 shocks decaying at the rate proportionate to 1/{t\sqrt{\log t}}. We address this conjecture by first identifying the asymptotic profile which exhibit 2 shocks, as a self-similar solution of a related Burgers equation, and then proving its stability and the conjectured rate of decay for general (non-spherically symmetric) perturbations. This is joint work with D. Ginsberg.
In 1945, Landau argued that spherically symmetric solutions which form weak shocks will settle to a profile with 2 shocks decaying at the rate proportionate to 1/{t\sqrt{\log t}}. We address this conjecture by first identifying the asymptotic profile which exhibit 2 shocks, as a self-similar solution of a related Burgers equation, and then proving its stability and the conjectured rate of decay for general (non-spherically symmetric) perturbations. This is joint work with D. Ginsberg.
11:30 AM - 12:15 PM EST
Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential
11th Floor Lecture Hall
Virtual Speaker
Dmitry Pelinovsky, McMaster University
Abstract
In order to prove the existence of a ground state (a positive, radially symmetric solution in the energy space), we develop the shooting method and deal with a one-parameter family of classical solutions to an initial-value problem for the stationary equation. We prove that the solution curve (the graph of the eigenvalue parameter versus the supremum norm) is oscillatory below a threshold and monotone above a threshold. Compared to the existing literature, rigorous asymptotics are derived by constructing families of solutions to the stationary equation with functional-analytic rather than geometric methods. The same analytical technique allows us to characterize the Morse index of the ground state.
12:30 - 2:00 PM EST
Lunch/Free Time
2:00 - 2:45 PM EST
Rigidity for solutions to the quintic NLS equation at the ground state level
11th Floor Lecture Hall
Benjamin Dodson, John Hopkins University
Abstract
In this talk, we will prove rigidity for solutions to the quintic nonlinear Schrodinger equation in one dimension, at the level of the ground state. Specifically, we show that the only solutions that fail to scatter are the solitons and the pseudoconformal transformation of the solitons.
3:00 - 4:30 PM EST
Lightning Talks followed by Coffee Break and discussions
Lightning Talks - 11th Floor Lecture Hall
Wednesday, December 08, 2021
9:00 - 9:45 AM EST
A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schrodinger Equation
11th Floor Lecture Hall
Virtual Speaker
Nataša Pavlović, University of Texas at Austin
Abstract
Many mathematical works have focused on understanding the manner in which the dynamics of the nonlinear Schrodinger equation (NLS) arises as an effective equation. By effective equation, we mean that solutions of the NLS equation approximate solutions to an underlying physical equation in some topology in a particular asymptotic regime. For example, the cubic NLS is an effective equation for a system of N bosons interacting pairwise via a delta or approximate delta potential. In this talk, we will advance a new perspective on deriving an effective equation, which focuses on structure. In particular, we will show how the Hamiltonian structure for the cubic NLS in any dimension arises from corresponding structure at the N-particle level. Also we will discuss what we have learned so far about understanding origins of integrability of the 1D cubic NLS. The talk is based on joint works with Dana Mendelson, Andrea Nahmod, Matthew Rosenzweig and Gigliola Staffilani.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Gibbs measures, canonical stochastic quantization, and singular stochastic wave equations
11th Floor Lecture Hall
Virtual Speaker
Tadahiro Oh, The University of Edinburgh
Abstract
In this talk, I will discuss the (non-)construction of the focusing Gibbs measures and the associated dynamical problems. This study was
initiated by Lebowitz, Rose, and Speer (1988) and continued by Bourgain (1994), Brydges-Slade (1996), and Carlen-Fröhlich-Lebowitz (2016). In the one-dimensional setting, we consider the mass-critical case, where a critical mass threshold is given by the mass of the ground state on the real line. In this case, I will show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988).
In the three dimensional-setting, I will first discuss the construction of the $\Phi^3_3$-measure with a cubic interaction potential. This problem turns out to be critical, exhibiting a phase transition: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. Then, I will discuss the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic $\Phi^3_3$-model). As for the local theory, I will describe the paracontrolled approach to study stochastic nonlinear wave equations, introduced in my work with Gubinelli and Koch (2018). In the globalization part, I introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.
The first part of the talk is based on a joint work with Philippe Sosoe (Cornell) and Leonardo Tolomeo (Bonn), while the second part is based on a joint work with Mamoru Okamoto (Osaka) and Leonardo Tolomeo (Bonn).
In the three dimensional-setting, I will first discuss the construction of the $\Phi^3_3$-measure with a cubic interaction potential. This problem turns out to be critical, exhibiting a phase transition: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. Then, I will discuss the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic $\Phi^3_3$-model). As for the local theory, I will describe the paracontrolled approach to study stochastic nonlinear wave equations, introduced in my work with Gubinelli and Koch (2018). In the globalization part, I introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.
The first part of the talk is based on a joint work with Philippe Sosoe (Cornell) and Leonardo Tolomeo (Bonn), while the second part is based on a joint work with Mamoru Okamoto (Osaka) and Leonardo Tolomeo (Bonn).
11:30 AM - 12:15 PM EST
Mathematical Construction for Gravitational Collapse
11th Floor Lecture Hall
Yan Guo, Brown University
Abstract
We will discuss recent constructions of blowup solutions for describing gravitational collapse for Euler-Poisson system.
12:30 - 12:40 PM EST
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:40 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Internal Modes and Radiation Damping for 3d Klein-Gordon equations
11th Floor Lecture Hall
Virtual Speaker
Fabio Pusateri, University of Toronto
Abstract
We consider quadratic Klein-Gordon equations with an external potential $V$ in $3+1$ space-time dimensions. We assume that $V$ is generic and decaying, and that the operator $-\Delta + V + m^2$ has an eigenvalue $\lambda^2 < m^2$. This is a so-called ‘internal mode’ and gives rise to time-periodic localized solutions of the linear flow. We address the question of whether such solutions persist under the full nonlinear flow. Our main result shows that small nonlinear solutions slowly decay as the energy is transferred from the internal mode to the continuous spectrum, provided a natural Fermi golden rule holds. Moreover, we obtain very precise asymptotic information including sharp rates of decay and the growth of weighted norms. These results extend the seminal work of Soffer-Weinstein for cubic nonlinearities to the case of any generic perturbation. This is joint work with Tristan Léger (Princeton University).
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
Solutions to the KdV and Related Equations With Almost Periodic Initial Data
11th Floor Lecture Hall
David Damanik, Rice University
Abstract
We discuss recent work concerning the existence, uniqueness, and structure of solutions to the KdV equation, as well as related ones, with almost periodic initial data. The talk is based on several joint works with a variety of co-authors, including Ilia Binder, Michael Goldstein, Yong Li, Milivoje Lukic, Alexander Volberg, Fei Xu, and Peter Yuditskii.
Thursday, December 09, 2021
9:00 - 9:45 AM EST
The Quartic Integrability and Long Time Existence of Steep Water Waves in 2D
11th Floor Lecture Hall
Virtual Speaker
Sijue Wu, University of Michigan
Abstract
Abstract. It is known since the work of Dyachenko & Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non trivial resonant manifold. In this talk, I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals Ej (t), directly in the physical space, which are explicit in the Riemann mapping variable and involve material derivatives of order j of the solutions for the 2d water wave equation, so that ddtEj (t) is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than ε, then the lifespan of the solution for the 2d water wave equation is at least of order O(ε−3), and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size ε, then the lifespan of the solution is at least of order O(ε−5/2). Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Global wellposedness of the Zakharov System below the ground state
11th Floor Lecture Hall
Virtual Speaker
Sebastian Herr, Bielefeld University
Abstract
We consider the Cauchy problem for the Zakharov system with a focus on the energy-critical dimension d = 4 and prove that global well-posedness holds 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 Schr ̈odinger equation with potentials solving the wave equation. This is joint work with Timothy Candy and Kenji Nakanishi.
11:30 AM - 12:15 PM EST
Stability of solitary waves in the generalized Benjamin-Ono equation and its 2d extensions
11th Floor Lecture Hall
Svetlana Roudenko, Florida International University
Abstract
We discuss solitary wave solutions in the generalized Benjamin-Ono equation, including the critical and supercritical cases. We then look at its higher-dimensional generalization in 2d, the Shrira equation, and its fractional generalization, the HBO (Higher-dimensional Benjamin-Ono) equation and examine the behavior of solutions in various cases as well as the stability of solitary waves.
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Low regularity solutions for nonlinear waves
11th Floor Lecture Hall
Daniel Tataru, University of California, Berkeley
Abstract
The sharp local well-posedness result for generic nonlinear wave equations was proved in my work with Smith about 20 years ago. Around the same time, it was conjectured that, for problems satisfying a suitable nonlinear null condition,
the local well-posedness threshold can be improved. In this talk, I will describe the first result establishing this conjecture for a good model. This is joint work with Albert Ai and Mihaela Ifrim.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
Simple motion of stretch-limited elastic strings
11th Floor Lecture Hall
Virtual Speaker
Casey Rodriguez, University of North Carolina
Abstract
Perfectly flexible strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (force produces stretching) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of force is applied the stretch of the string is maximized (becoming inextensible). In this talk, we discuss a simple model for these stretch-limited elastic strings, in what way they model ``elastic" behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.
Friday, December 10, 2021
9:00 - 9:45 AM EST
Kink stability in nonlinear Klein-Gordon equations
11th Floor Lecture Hall
Pierre Germain, NYU - Courant Institute
Abstract
Nonlinear Klein-Gordon equations whose potential have a double well admit kink solutions (most famous examples: Phi4, Sine-Gordon). I will present joint work with Fabio Pusateri which establishes the stability of this kink under some spectral conditions on the linearized problem. The key idea of the proof is to view the problem through the distorted Fourier transform associated with the linearized problem.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Non-trivial self-similar blowup in energy supercritical wave equations
11th Floor Lecture Hall
Birgit Schoerkhuber, University of Innsbruck, Austria
Abstract
Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new self-similar solutions with non-trivial profiles, which are completely explicit in all supercritical dimensions. Furthermore, we analyse their stability locally in backward light cones without symmetry assumptions. This involves a delicate spectral problem that we are able to solve rigorously only in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations in a backward light cone of the blowup point. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck).
11:30 AM - 12:15 PM EST
The stability of charged black holes
11th Floor Lecture Hall
Virtual Speaker
Elena Giorgi, Columbia University
Abstract
Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
12:30 - 2:00 PM EST
Lunch/Free Time
2:00 - 2:45 PM EST
Invariance of the Gibbs measures for the periodic generalized KdV equations
11th Floor Lecture Hall
Virtual Speaker
Andreia Chapouto, UCLA
Abstract
In this talk, we consider the periodic generalized Korteweg-de Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing local-in-time dynamics in the support of the measure. Since gKdV is analytically ill-posed in the L2-based Sobolev support, we instead prove deterministic local well-posedness in some Fourier-Lebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Once we construct local-in-time dynamics, we apply Bourgain's invariant measure argument to prove almost sure global well-posedness of the defocusing gKdV and invariance of the Gibbs measure. Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations. This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).
3:00 - 3:45 PM EST
Global in x Stability of Prandtl's Boundary Layer for 2D, Stationary Navier-Stokes Flows
11th Floor Lecture Hall
Virtual Speaker
Sameer Iyer, UC Davis
Abstract
In this talk, I will discuss a recent work which proves stability of Prandtl's boundary layer in the vanishing viscosity limit. The result is an asymptotic stability result of the background profile in two senses: asymptotic as the viscosity tends to zero and asymptotic as x (which acts a time variable) goes to infinity. In particular, this confirms the lack of the "boundary layer separation" in certain regimes which have been predicted to be stable. This is joint work w. Nader Masmoudi (Courant Institute, NYU).
4:00 - 4:30 PM EST
Coffee Break
11th Floor Collaborative Space
