This workshop is centered around wave propagation in complex environnements, and how its dynamics may be related to both the medium and nonlinear effets. It brings together people working at the crossroads between dynamical systems, nonlinear PDEs, control theory, numerical analysis and microlocal analysis.

**Mini -Courses **

There will be 4 mini-courses by **Pierre Raphae****l **and **Jérémie Szeftel. **

**Finite time blow-up for compressible fluids and supercritical defocusing NLS**

Lecture 1: Introduction to blow up analysis

We will introduce the basic ingredients of finite time blow-up analysis for nonlinear evolution PDE and illustrate them on the concrete example of focusing NLS.

Lecture 2: Blow up for compressible fluids and supercritical defocusing NLS

We will present our main results on finite time blow up for compressible fluids and supercritical defocusing NLS.

Lecture 3: Font renormalization and profile construction

We will illustrate the notion of front renormalization on the concrete example of the nonlinear heat equation. We will then show how it can be used to link our result on finite time blow up for supercritical defocusing NLS to the construction of self similar profiles for compressible Euler. We will finally discuss the construction of such profiles.

Lecture 4: Nonlinear stability

Starting from our construction of self similar profiles for compressible Euler we will outline the proof of the nonlinear stability leading to finite time blow up for compressible fluids and supercritical defocusing NLS.

**Talks **

**Thomas Alazard**

*The virial theorem for water waves*

This talk is about a joint work with Claude Zuily, where we establish several identities related to the equipartition of energy for water waves.

**Corentin Audiard **

*Nonlinear stability of periodic waves for a KdV type equation in large dimension*

We consider a PDE which is a toy model for the study of stability of periodic waves of (g)KdV under transverse dispersive perturbation. In dimension 1 the existence of small amplitude periodic waves (close to a constant state) is obtained, and we give a characterization of their linear stability in L^{2 }(**R**) with a Benjamin-Feir type index. Nonlinear stability is then obtained when the dimension is large enough by somewhat classical arguments. Some (large) room for improvements should also be discussed.

This is joint work with M. Rodriguez and C.Sun.

**Dario Bambusi**

*Almost global existence for Hamiltonian PDEs on general flat tori*

We will present a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential and a beam equation. The abstract theorem can also be used to ensure effective stability of the plane waves in NLS. The main technical novelty is that we deal with general tori, including irrational tori, in which the differences of the eigenvalues of the Laplacian are dense on the real axe.

Joint work with R. Feola and R. Montalto.

**Valeria Banica**

*Blow-up for the 1D cubic NLS*

We consider the cubic 1D NLS on **R** and prove a blow-up result for functions that are of borderline regularity. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite time. This functional framework allows, after using a pseudo-conformal transformation, to reduce the problem to a large time study of a periodic Schrödinger equation with non-autonomous cubic nonlinearity. The blow-up result corresponds to a long range asymptotic completeness result for the new equation.

Finally, as an application we give conditions on curvature and torsion of a smooth curve to insure the existence of a binormal flow solution that generates several singularities in finite time.

This is a joint work with Renato Lucà, Nikolay Tzvetkov and Luis Vega.

**Jacopo Bellazini**

*Ground state energy threshold and blow-up for NLS with competing nonlinearities*

The aim of the talk is to discuss qualitative properties of the nonlinear Schrödinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. Fixed the mass of the problem, we completely answer the question wether the ground state energy , which is a threshold between global existence and formation of singularities, is achieved. As a byproduct of the variational characterization of the ground state energy, we show the existence of blowing-up solutions in finite time, for any initial data with energy below the ground state energy threshold in case of cylindrical symmetry.

**Joackim Bernier **

*Typical dynamics of some resonant Hamiltonian PDEs*

We will discuss some results (some past and some others in preparation) about long time stability of small typical solutions of some resonant Hamiltonian PDEs.

**Massimilano Berti**

*Hamiltonian Birkhoff normal form for water waves with constant vorticity : almost global existence*

We will present an almost global in time existence result of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth and any surface tension belonging to a full measure set. The proof demands a Hamiltonian paradifferential Birkhoff normal form reduction for quasi-linear PDEs in presence of resonant wave interactions: the normal form may be not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that usual paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) does not preserve the Hamiltonian structure. A major novelty of this paper is to develop an algorithmic perturbative procedure a’ la Darboux to correct usual paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. The symplectic correctors turn out to be smoothing perturbations of the identity, and therefore only slightly modify the paradifferential structure of the equations. The Darboux procedure which recovers the nonlinear Hamiltonian structure is written in an abstract functional setting, in order to be applicable also in other contexts. Joint work with A. Maspero and F. Murgante.

**Lucrezia Cossetti**

*A limiting absorption principle for time-harmonic isotropic Maxwell and Dirac equation*

In this talk we investigate the L^{p} − L^{q} mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell and perturbed Dirac operator. As spectral parameters close to the spectrum are also covered by our analysis, we establish a L^{p} – L^{q} type limiting absorption principle for these operators. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations.

The talk is based on a joint work with R. Mandel and on an ongoing project with R. Mandel and R. Schippa.

**Benoit Grébert**

*Dynamics of quintic nonlinear Schrödinger equations in H ^{{2/5}}(T)*

We will start this talk by recalling some recent results I obtained with J. Bernier on the Birkhoff normal form with low regularity. Then we will focus on an even more recent result with J. Bernier and T. Robert : in order to further lower the regularity of solutions, we succeed in integrating Strichartz estimates (encoding the dispersive effects of the equations) in Birkhoff normal form techniques. As a consequence, we deduce a result on the long time behavior of quintic NLS solutions on the circle for small but very irregular initial data. Note that, since 2/5<1, we cannot claim conservation of energy and more importantly, since 2/5<1/2, we must dispense with the algebra property of H^{s}. This is the first dynamical result where we use the dispersive properties of NLS in a context of Birkhoff normal form.

**Masayuki Hayashi **

*Instability of stationary solutions for double power nonlinear Schrödinger equations*

We consider the nonlinear Schrödinger equation with doubler power nonlinearities in L^{2 } – subcritical setting. The equation has algebraically decaying stationary solution as well as usual standing waves decaying exponentially with positive frequency. Stability/instability of the stationary solution is outside the framework of general abstract theory and one of the difficulties come from the lack of coercivity properties of the linearized operator. In this talk we study instability of the stationary solution.

**Beatrice Langella **

*Reducibility and nonlinear stability for a quasi-periodically forced NLS on the torus*

Motivated by the problem of long time stability vs instability of KAM tori of the nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus **T**^{2}**,** in this talk we will consider a quasi-periodically forced NLS equation on **T**^{2}, arising from the linearization of the NLS at a KAM torus. For such equation we will show how to prove a reducibility result, as well as long time stability of the origin. A fundamental step in the proof is to obtain a refined asymptotic expansion of the frequencies of the linearized operator, which allows to impose Melnikov conditions at any arbitrary order.

This talk is based on a joint work with Emanuele Haus, Alberto Maspero, and Michela Procesi.

**Camille Laurent**

*Observability for systems of (non)linear waves*

In this talk, we will report on some results about the control of systems of wave equations coupled by lower orders terms. We will describe the link with an ODE system along the geodesics. Then, we will discuss some related results of stabilization for nonlinear systems of waves.

This is joint work with with Yan Cui and Zhiqiang Wang for the linear part and with Radhia Ayechi and Moez Khenissi and for the nonlinear part

**Zexing Li **

*On asymptotic stability and Strichartz estimate for self-similar blowup of mass supercritical NLS*

For slightly mass supercritical nonlinear Schrodinger equations (NLS), self-similar blowup has been proven to exist and generate stable blowup dynamics. However, the asymptotic stability was missing. With suitable self-similar profiles constructed recently, we take one further step to show their finite codimensional asymptotic stability. One core ingredient is a Strichartz estimate for the linearized matrix operator, where an “enhanced dispersion” phenomenon for the propagator is exploited.

**Laurent Michel**

*Sharp asymptotics for the exit time of non reversible stochastic processes*

We consider the problem of the exit time of an open set for metastable non-reversible stochastic processes. A correspondence is established between the expectation of the exit time and the inverse of the principal eigenvalue of the generator. In a suitable geometric framework, we then prove an Eyring-Kramers formula for this eigenvalue. Joint work with D. Le Peutrec and B. Nectoux

**Francesca Prinari**

*Sobolev embedding and distance functions*

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D^{{1,p}}_{0} into L^{q} and the summability properties of the distance function. We prove that in the superconformal case (i.e. when p is larger than the dimension) these two facts are equivalent, while in the subconformal and conformal cases (i.e. when p is less than or equal to the dimension) we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behaviour of the positive solution of the Lane-Emden equation for the p-Laplacian with sub-homogeneous right-hand side, as the exponent p diverges to + infinity (a joint work with L. Brasco and A.C. Zagati).