Workshop

This talk will be divided into two parts.

The first part will be devoted to the introduction of the cubic Szegö equation on the circle and of its integrable structure, with applications to existence of solutions displaying wave turbulence.

The second part will give an idea of  works in progress about extensions of wave turbulence to  related Hamiltonian PDEs.

In this talk, we consider kinetic equations containing random
terms. The kinetic models contain a small parameter and it is well
known that, after scaling, when this parameter goes to zero the limit
problem is a diffusion equation in the PDE sense, ie a parabolic equation
of second order. A smooth noise is added, accounting for external perturbation.
It scales also with the small parameter. It is expected that the limit
equation is then a stochastic parabolic equation where the noise is in
Stratonovitch form.
Our aim is to justify in this way several SPDEs commonly used.
We first treat linear equations with multiplicative noise. Then show how
to extend the methods to nonlinear equations or to the more physical
case of a random forcing term.
The results have been obtained jointly with S. De Moor and J. Vovelle

We prove that the flow of the mass-critical NLS in two dimensions cannot squeeze a ball in $L^2$ into a cylinder of lesser radius.  This is a PDE analogue of Gromov's non-squeezing theorem for an infinite-dimensional Hamiltonian PDE in infinite volume.  This is joint work with R. Killip and X. Zhang

. I will present some recent optimal regularity results for parabolic SPDEs driven by space-time white noise.

Our results are connected closely to questions about how sensitive the solution to a parabolic SPDE is to small changes in the initial data.

This is based on joint work with Le Chen and Kunwoo Kim

We survey recent progress on asymptotic fluctuations in the KPZ universality class

In this talk, I will present our recent work on the local in time existence of two-dimensional gravity water waves with angled crests. Specifically,  we construct an energy functional $E(t)$ that allows for angled crests in the interface. We show that for any initial data satisfying $E(0)<\infty$, there is $T>0$, depending only on $E(0)$, such that the water wave system is solvable for time $t\in [0, T]$. Furthermore we show that for any smooth initial data, the unique solution of the 2d water wave system remains smooth so long as $E(t)$ remains finite.

The point of this talk is to show how certain well-posedness results  that are not available using deterministic techniques involving Fourier and harmonic analysis 

can be obtained when introducing randomization in the set of initial data. Along the way I will also prove a certain “probabilistic propagation of regularity” for certain almost sure globally well-posed dispersive equations. This talk is based on joint work with A. Nahmod

We introduce the theory of wave turbulence as a systematic approach to studying the energy distribution across scales in nonlinear dispersive PDE. This is done by deriving an effective equation for the energy density of the system in a statistical setting, by taking weak nonlinearity and infinite volume limits. The resulting equation is called the “wave kinetic equation”, and it gives, at a formal level, a lot of insight into the out-of-equilibrium dynamics and statistics of nonlinear dispersive systems. The fundamental problem here is to give a rigorous derivation of this formally derived equation. Without any stochastic element in the system, such problems are often too difficult to resolve (even in much simpler ODE settings). We will show how this equation can be derived starting from the nonlinear Schrodinger equation on a large torus, in the presence of an appropriate random force, and in the weakly nonlinear infinite volume limit. This is joint work with Isabelle Gallagher and Pierre Germain

I will try to explain, starting from the historical papers, what has been understood (and not understood) from a rigorous point of view, about classical Hamiltonian systems. For example a chain of massive bodies, or a coupled set of rotators, which are driven out of equilibrium by stochastic forces acting on the ends of the chain

Water waves are disturbances of the free surface of a liquid. They are, in general, produced

by the immersion of a solid body or by impulsive pressures applied on the free surface. The 

two questions we discuss in this talk are the following: 

- which waves can be generated by blowing on a localized portion of the free surface ? 

Our main result with Pietro Baldi and Daniel Han-Kwan asserts that one can generate any small amplitude, periodic in x, two-dimensional, gravity-capillary water waves. 

- consider now gravity water waves in a rectangular tank. Is-it possible to estimate their energy by

looking only at the motion of some of the curves of contact between the free surface and the vertical walls

We consider a class of nonlinear mappings F

We will review some recent results on the stochastic Landau-Lifshitz equation which models temperature effects on magnetization dynamics in micro-magnetism. The particularity of the model is the geometric constraint the magnetization is a unit vector — which makes the equation nonlinear. The associated stochastic partial differential equation, taking account of temperature effects, has known some recent improvements, from the point of view of mathematical analysis as well as numerical analysis, but still offers a lot of open problems

We will review some of the work conducted over the past five years
on the asymptotic behavior of solutions to dispersive nonlinear evolution equations

We describe a general approach to the renormalisation step in the theory of regularity structures based on Hopf algebras of labelled trees (joint work with Yvain Bruned and Martin Hairer)."

In this talk (essentially based in a joint work with Tristan Roy), I will review some recent results about the blow-up of the scale-invariant Sobolev norm for nonlinear wave equations in space dimension 3.

We consider the question of stabilization for the damped wave equation on tori
$$(\partial_t^2 -\Delta )u +a(x) \partial _t u =0.$$
When the damping coefficient $a(x)$ is continuous the question is quite well understood and the geometric control condition is necessary and sufficient for uniform (hence exponential) decay to hold. When $a(x)$ is only $L^{\infty}$ there are still gaps in the understanding.
Using second microlocalization we completely solve the question for 
Damping coefficients of the form
$$a(x)=\sum_{i=1}^{J} a_j 1_{x\in R_j},$$
Where $R_j$ are cubes.
This is a joint work with P. Gérard

The prototypical problem of data assimilation is weather forecasting. The goal is to provide state predictions for nonlinear and extremely high dimensional models, the novelty is to determine how best to guide these predictions using the now vast amounts of observational data that is available to forecasters. Most data assimilation algorithms have been developed within the applied sciences, notably in meteorology. As a consequence these algorithms sit on little to no mathematical foundation, instead they are ad hoc methods whereby well-understood algorithms are tweaked to make them more suitable to the given problem. Nevertheless, numerical evidence suggests that these algorithms work - they often possess mathematically well understood properties, such as filter accuracy (tracking the right signal) and stability (ergodicity).

We will discuss two roles that mathematicians can play in this field: 1 - to develop understanding of the mathematics behind seemingly ad hoc forecasting algorithms and 2 - to use this understanding to propose new algorithms that are strongly backed by theory. 

This is based on several joint works with Andy Majda, Andrew Stuart and Xin Tong.