# Schedule of the Workshop "Harmonic Analysis Methods in Dispersive PDEs"

## Tuesday, June 10

 9:00 - 10:00 Joachim Krieger: Conditional Stability of the Catenoid under Vanishing Mean Curvature Flow 10:00 - 11:00 Jeremy Marzuola: Quasilinear Schrödinger equations 11:00 - 11:30 Coffee break 11:30 - 12:30 Ronald Donninger: Self-similar blowup in supercritical wave equations 12:30 - 14:00 Lunch break 14:00 - 15:00 Alexandru Ionescu: On the global stability of the wave map equation in Kerr spaces 15:00 - 16:00 Luis Vega: The dynamics of vortex filaments with corners 16:00 - 16:30 Tea and cake

## Wednesday, June 11

 9:00 - 10:00 Nikolay Tzvetkov: Growing Sobolev norms for the cubic defocusing Schroedinger equation 10:00 - 11:00 Hani Zaher: Effective dynamics for the cubic nonlinear Schroedinger equation confined by domain or potential 11:00 - 11:30 Coffee break 11:30 - 12:30 Junfeng Li: The Global well-posedness for 3 D Kadomtsev-Petviashvili II problem 12:30 - 14:00 Lunch break 14:00 - 15:00 Stefan Steinerberger: Convolution estimates and global nonlinear Brascamp-Lieb inequalities 15:00 - 16:00 Sebastian Herr: The cubic Dirac equation in the critical Sobolev space 16:00 - 16:30 Tea and cake

## Thursday, June 12

 9:00 - 10:00 Benoit Pausader: On the Euler-Maxwell system for electrons 10:00 - 11:00 Mihaela Ifrim: Two dimensional water waves in holomorphic coordinates 11:00 - 11:30 Coffee break 11:30 - 12:30 Jean Marc Delort: Semiclassical microlocal normal forms and global solutions of modified one-dimensional Klein-Gordon equations 12:30 - 14:00 Lunch break 14:00 - Excursion

## Friday, June 13

 9:00 - 10:00 Gigliola Stafillani: The periodic Schrodinger equation and its solutions with mass or infinite energy 10:00 - 11:00 Dirk Hundertmark: Some mathematical challenges from non-linear fiber optics 11:00 - 11:30 Coffee break 11:30 - 12:30 Nils Strunk: The energy-critical NLS posed on compact 3-manifolds 12:30 - 14:00 Lunch break 14:00 - 15:00 Pedro Pérez Caro: On uniqueness of an inverse problem for Maxwell equations 15:00 - 16:00 Aynur Bulut: Probabilistic local and global well-posedness results for symmetric solutions of the nonlinear wave equation 16:00 - 16:30 Tea and cake

# Abstracts

(Underlined titles can be clicked for the video recording)

## Aynur Bulut: Probabilistic local and global well-posedness results for symmetric solutions of the nonlinear wave equation

In this talk, we will discuss recent work on new global well-posedness results for the nonlinear wave equation posed on the unit ball of $\mathbb{R}^3$ (with radial symmetry and zero boundary conditions) and on the "periodic cylinder" $B_2{T}$ (with radiality and zero boundary conditions in the $B_2$ variable). We work with supercritical initial data distributed as a Gaussian random process. In particular, our data belongs a.s. to the ill-posed regime for the initial value problem, and probabilistic considerations are therefore an essential component of our analysis. Important ingredients in the local analysis include the derivation of probabilistic a priori bounds for the relevant evolutions (making use of the product structure of the domain in the cylinder setting). To extend the local results globally in time, our analysis in the setting of the ball is based on invariance properties of the Gibbs measure, while for the cylinder we appeal to an instance of the high-low method of J. Bourgain. Our results on the cylinder can be viewed as an interpolation between corresponding results on the 3D torus and the 3D ball.

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## Pedro Pérez Caro: On uniqueness of an inverse problem for Maxwell equations

In this talk I will show a uniqueness result for an inverse boundary value problem arising in electrodynamics, in the context of the time-harmonic Maxwell equations. This result was proven in a joint paper with Ting Zhou where we assumed the electromagnetic properties of the medium, namely the magnetic permeability, the electric permittivity and the conductivity, to be described by continuously differentiable functions. Our result extends a theorem by Haberman and Tataru for the Calderón problem.

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## Jean Marc Delort: Semiclassical microlocal normal forms and global solutions of modified one-dimensional Klein-Gordon equations

he method of Klainerman vector fields plays an essential role in the study of global existence of solutions of nonlinear hyperbolic PDEs, with small, smooth, decaying Cauchy data. Nevertheless, it turns out that some equations of physics, like the one dimensional water waves equation with finite depth, do not possess any Klainerman vector field. The goal of this talk is to explain, on a model equation, a substitute to the Klainerman vector fields method, that allows one to get global existence results, even in the critical case for which linear scattering does not hold at infinity. The main idea is to use semiclassical pseudodifferential operators instead of vector fields, combined with microlocal normal forms, to reduce the nonlinearity to expressions for which a Leibniz rule holds for these operators.

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## Ronald Donninger: Self-similar blowup in supercritical wave equations

I discuss a general perturbative method to study the stability of self-similar solutions to nonlinear wave equations. The approach applies to energy-supercritical problems. The talk is based on joint work with Birgit Schörkhuber from Vienna.

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## Dirk Hundertmark: Some mathematical challenges from non-linear fiber optics

We describe some recent rigorous work on soliton-like pulses in dispersion managed optical fiber channels:

"Dispersion management'' refers to the engineering of an optical fiber channel with alternating spans of positive (normal) and negative (anomalous) dispersion fiber (periodic or otherwise) in order to achieve greater stability, bandwidth etc of optical information transfer. This technology has lead to a 100fold increase in bandwidth in long-haul optical transmission lines. The simplest mathematical model describing pulses in a glass-fiber cable is the scalar one-dimensional nonlinear Schroedinger equation with cubic nonlinearity. "Dispersion management'' means that the coefficient of dispersion is a function of distance (e.g. periodic) along the fiber waveguide.

To model dispersion managed fiber channels one also averages over one period, yielding the Gabitov-Turitsyn equation, which is a non-local version of the non-linear Schroedinger equation.

It is well known that with constant negative (i.e., anomalous) dispersion there are soliton-like localized solutions and, not much of a surprise, for dispersion managed systems if the dispersion is, on the average, anomalous then there are again stable solitons.

However, in physical experiments, as well as numerical studies, it has long been observed that one gets soliton-like localized solutions even for average dispersion equal to **zero** dispersion. This was a surprise, both physically and mathematically, because the conventional wisdom had been that solitons emerge from a combination of nontrivial linear dispersion and nonlinearity. Something more subtle is going on in the zero average dispersion case, which is also the most important case from an applications point of view.

Rigorous results on soliton-like pulses for the Gabtov-Turitsyn equation, the so-called dispersion management solitons, have been rare (I know of 7 or so), which is mainly due to its non-locality, which makes it hard to study. Rigorous results for zero average dispersion are even rarer, since this case is a singular limit. This is quite in contrast to the enormous amount of experimental, numerical and theoretical work (try to google "dispersion management'').

We will discuss recent work on the decay and regularity properties of dispersion management solitons. Our results include a simple proof of existence of solutions of the dispersion management equation under mild conditions on the dispersion profile, which includes all physically relevant cases, regularity of weak solutions, and most recently a proof of exponential decay of dispersion management solitons, which confirms the theoretically and experimentally seen fact that dispersion management solitons are very well-localized. This is joint work with Burak Erdoğan, William Green, and Young-Ran Lee

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## Sebastian Herr: The cubic Dirac equation in the critical Sobolev space

In this talk we will consider the initial value problem associated with the cubic Dirac equation. We will discuss its well-posedness for small initial data in the critical space $H ^1$${R}^3$. This is joint work with Ioan Bejenaru.

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## Alexandru Ionescu: On the global stability of the wave map equation in Kerr spaces

On the global stability of the wave map equation in Kerr spaces Abstract: I will discuss some recent work (joint with S. Klainerman) on the global stability of a stationary axially-symmetric solution of the wave map equation in Kerr spaces of small angular momentum.

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## Junfeng Li: The Global well-posedness for 3 D Kadomtsev-Petviashvili II problem

In this talk, I will present the global well posed results for the 3D Kadomtsev-Petviashvili II problem in some scaling critical spaces. To achieve this result, a new bilinear estimate is set up. And some fundamental symmetrie is used.

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## Jeremy Marzuola: Quasilinear Schrödinger equations

We survey some recent results by the speaker, Jason Metcalfe and Daniel Tataru for small data local well-posedness of quasilinear Schrödinger equations. In addition, we will discuss some applications recently explored with Jianfeng Lu and potentially partial progress towards the large data short time problem.

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## Stefan Steinerberger: Convolution estimates and global nonlinear Brascamp-Lieb inequalities

Bilinear estimates for dispersive equations translate into convolution estimates for measures supported on the characteristic hypersurfaces. Convolution estimates are of interest in themselves and have been studied by Bennett-Carbery-Wright, Bennett-Bez, Bejenaru-Herr-Tataru, Bejenaru-Herr and others. We give a new and simple approach allowing us to improve on earlier results and derive a global convolution estimates for measures supported on transversal hypersurfaces. The same idea allows us to derive global, nonlinear Brascamp-Lieb inequalities. This is joint work with Herbert Koch.

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## Nikolay Tzvetkov: Growing Sobolev norms for the cubic defocusing Schroedinger equation

The goal of the talk is to describe recent results of Zaher Hani, Benoit Pausader, Nicola Visciglia and the speaker on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.

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## Luis Vega: The dynamics of vortex filaments with corners

I shall present some recent work about the evolution of vortex filaments that follow the geometric law of the binormal: a point of the filament moves in the direction of the binormal with a speed that is proportional to the curvature. First I'll consider the case of a curve (filament) that is regular except at a point where it has a corner (joint work with V. Banica). Then, I'll look at the case of a regular polygon (joint work with F. de la Hoz). In this latter situation the dynamics is rather complicated and can be seen as a non-linear Talbot effect. Connection with the multifractality exhibited by the graph of the so-called Riemann non-differentiable function will be made.

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## Hani Zaher: Effective dynamics for the cubic nonlinear Schroedinger equation confined by domain or potential

We start by considering the cubic nonlinear Schroedinger (NLS) equation confined in a 2D box of size L. By taking the large box limit in a particular regime of small data, we derive a new equation that describes the effective dynamics of NLS over long nonlinear time scales. The obtained equation is called the "continuous resonant" (CR) equation and can also be interpreted as the equation for high frequency envelopes of NLS on the unit 2-torus. The (CR) equation turns out to satisfy rather surprising properties and symmetries, like leaving the Fourier transform and the eigenspaces of the quantum harmonic oscillator invariant by its flow. This signals to a relationship between the (CR) equation and the NLS equation on Rd with harmonic trapping. Indeed, we will see how the dynamics of (CR) can be used to describe the long-time behavior of NLS in both isotropic and non-isotropic (cigar-shaped) harmonic traps. I will present works in collaboration with E. Faou, P. Germain, and L. Thomann.

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