• Harmonic Analysis and Partial Differential Equations
• Workshop: Harmonic Analysis Methods in Dispersive PDEs

• Schedule
• Participants

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 (with radial symmetry and zero boundary conditions) and on the "periodic cylinder" (with radiality and zero boundary conditions in the 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.

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.

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.

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.

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 . This is joint work with Ioan Bejenaru.

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.

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.

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.

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.

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.

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.

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 R^d 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.