The goal of this talk is to study the generation and the damping (or stabilization) of water waves in a tank. We consider the cases with and without surface tension and consider two very different approaches. Firstly, I will explain that one can generate any 2D gravity-capillary water waves by blowing only on a localized portion of the free surface (joint work with Pietro Baldi and Daniel Han-Kwan). Secondly, I would like to explain how to obtain observability inequalities using only global quantities (instead of microlocal). Namely, in the second talk we will see how to use the multiplier method and the hamiltonian structure of the equation to obtain observability inequalities.

We prove that the KdV equation on the circle remains  exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders.
We use a procedure of reduction to constant coefficients up to order zero, classical Ingham inenquality and HUM method to prove the controllability of the linearized operator. Then a fine version of the Nash-Moser implicit function theorem due to Hörmander can be applied as a “black box”.

We will present the main idea of the paracontrolled calculus, which was recently introduced in the Euclidean situation by Gubinelli, Imkeller and Perkowski.
This gives an alternative approach to Hairer’s theory in order to deal with singular PDEs. We will then explain how we can extend it in many various situations, given by a heat semigroup with gradient estimates and describe how one can use it for solving the 2-3D Parabolic Anderson Model and the 3D stochastic Burger equations. This work is joint with Ismael Bailleul and Dorothee Frey.

In this talk I will present some uniqueness and stability results for an inverse boundary value problem consisting of recovering the conductivity of a medium from boundary measurements. This inverse problem was proposed by Calderón in 1980 and is the mathematical model for a medical imaging technique called Electrical Impedance Tomography which has promising applications in monitoring lung functions and as an alternative/complementary technique to mammography and Magnetic Resonance Imaging for breast cancer detection.  Since in real applications, the medium to be imaged may present quite rough electrical properties, it seems of capital relevance to know what are the minimal regularity assumptions on the conductivity to ensure the unique determination of the conductivity from the boundary measurements. This question is challenging and has been brought to the attention of many analysts. The results I will present provide uniqueness for Lipschitz conductivities and stability for continuously differentiable conductivities. These results were proved in collaboration with Keith Rogers, and Andoni García and Juan Reyes, respectively.

In this mini course I will present results on the reducibility of linear PDEs with quasi periodic in time potential. I will discuss the dynamical consequences of such a result: the control of the Sobolev norms of the solutions for all time.

In the second part I will focus on a recent result (in collaboration with E.Paturel) where we prove that a linear d-dimensional Schrödinger equation on R^d with harmonic potential and small t-quasiperiodic potential reduces to an autonomous system for most values of the frequency vector.

For Dirac and Klein-Gordon equations we will discuss new endpoint and multilinear Strichartz estimates. As an application, we present critical well-posedness and scattering results for the cubic Dirac equation in dimension 2 and 3 with small initial data in the critical space. This is joint work with Ioan Bejenaru (UC San Diego).

It is well known that the 3D Couette flow is stable for the Navier-Stokes dynamics. However, the main question is to find the size of the allowed perturbation depending on the viscosity. In these talks, we will discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 2D and 3D incompressible Navier-Stokes equations at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions below this threshold. The primary stability mechanisms are an anisotropic enhanced dissipation effect and an inviscid damping effect of the velocity component normal to the shear, both a result of the mixing caused by the large mean shear. After detailing these linear effects, we will discuss some of the important steps in the proof, such as the analysis of the weakly nonlinear (potential) instabilities connected to the non-normal nature of the linearization. Joint work with Jacob Bedrossian and Vlad Vicol and joint work with Jacob Bedrossian and Pierre Germain.