ERC Starting grant – Waves propagation in real life physics occur in media which are neither homogeneous or spatially infinity. The birth of radar/sonar technologies (and the raise of computed tomography) greatly motivated numerous developments in microlocal analysis and the linear theory. Only recently toy nonlinear models have been studied on a curved background, sometimes compact or rough. Understanding how to extend such tools, dealing with wave dispersion or focusing, will allow us to significantly progress in our mathematical understanding of physically relevant models. There, boundaries appear naturally and most earlier developments related to propagation of singularities in this context have limited scope with respect to crucial dispersive effects. Despite great progress over the last decade, driven by the study of quasilinear equations, our knowledge is still very limited. Going beyond this recent activity requires new tools whose development is at the heart of this proposal, including good approximate solutions going over arbitrarily large numbers of caustics, sharp pointwise bounds on Green functions, development of efficient wave packets methods, quantitative refinements of propagation of singularities (with direct applications in control theory), only to name a few important ones.
Seventh Itinerant Meeting in PDEs
Seventh Itinerant Meeting in PDE ‘s
from January 20 to January 22, 2016 at Parc Valorse, Nice
This year, the seventh edition of the Itinerant Meeting in PDEs takes place at Nice, France. The previous editions were celebrated in Trieste, Pisa, Rome, Bilbao and Bayonne (twice).
M. Berti (SISSA), L. Fanelli (Rome I), D. Lannes (Bordeaux),
F. Planchon (Nice), F. Rousset (Paris XI), L. Vega (BCAM Bilbao), N. Visciglia (Pise)
Local organising committee
F. Planchon (Nice), O. Ivanovici (Nice), A. Guitard (Nice)
Title and abstract
Mini-course : Control of water waves
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.
Exact controllability for quasi-linear perturbations of KdV
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”.
Paracontrolled calculus and singular PDEs (PAM and stochastic Burgers) through semigroup
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.
The inverse Calderón problem with Lipschitz conductivities
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.
Anne-Sophie DE SUZZONI
On large systems of fermions
I will present an equation on random variables which can in some sense be reduced to the Hartree-Fock equation. I will explain the relations between the two equations. Then, based on the similarity between the equation on random variables and the cubic nonlinear Schrödinger equation, I will give some of its properties, including global well-posedness in the energy space on the Euclidean spaces, torus and spheres of dimensions 2 and 3 in the defocussing case. I will finally come back to large systems of fermions by interpretating the results in terms of density operators.
Mini-course: On reducibility of linear PDEs with quasi periodic in time potential
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.
Endpoint Strichartz estimates and applications to cubic Dirac equations
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).
Mini-course: Mixing and enhanced dissipation in the inviscid limit of the Navier-Stokes equations near the 2D and 3D Couette flows
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.
How do you get here ?
Seventh Itinerant Meeting in PDE ‘s
Conference location:Théatre du Grand ChateauUniversité Nice Sophia-AntipolisParc Valrose 06108 Nice
From railway station (“gare Thiers”) to campus (“Valrose”), you can go: – by foot, approximately 30 minutes, – by tramway (light rail).
In both cases, you have to exit the railway station, take “avenue Thiers” on the left, to the tramway stop.
If you choose the tramway option, direction “Las Planas”, and exit at stop “Valrose Université” (3rd stop). If you choose the foot option, turn left, and follow the tramway line (avenue Malaussenta, and then avenue Borriglione). Once at “Valrose Université” stop, turn to the right, and then go straight to “avenue Joseph Vallot”. The campus is at the end of this street. Map of the tramway line: http://tramway.nice.fr/La-ligne-1/Le-trace
From airport to university, several options:
There is a free (blue) shuttle from Terminal 2 to Terminal 1. From Terminal 1, take bus 23 (ticket = 1 euro), direction “Vallon des Fleurs”. Exit at “Joseph Vallot”, go back and turn left into “Joseph Vallot” street. The campus “Valrose” is at the end of this street. It takes 45 minutes – 1 hour.
Faster: from terminal 1 or terminal 2 are 2 fast shuttles:
By bus 98: (ticket = 4 euros), direction “Riquier”, and exit at “Cathédrale Vieille Ville” stop. With the same ticket, take the tramway direction “Las Planas” and exit at “Valrose University” stop. Turn right into avenue “Joseph Vallot”.
By bus 99: take the shuttle until the railway station. Then use the above informations.
Taxi: much faster, much more expensive (approx. 30 euros, 40 at night or week-end, even more if you look like a tourist…). Don’t forget to mention “Valrose” campus, as there are several university campus in the city.
For those of you who will spend more time in Nice :
Some great art museums you SHOULD visit (both in Nice and the surrounding Riviera towns)