Thomas Alazard - Stabilization of the water-waves equations
Consider a fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of
gravity and surface tension. In this talk we explain that one can estimate its energy by looking only at a small localized portion of the free surface. This is used to study the damping of the energy by an absorbing beach where the water-wave energy is dissipated by using only the variations of the external pressure over a localized portion of the free surface. The analysis relies on the multiplier technique, the Craig-Sulem-Zakharov formulation of the water-wave problem, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations inspired by the analysis done by Benjamin and Olver of the conservation laws for water waves.
Diogo Arsénio - The vortex method for 2D ideal flows in exterior domains
The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation around the obstacle. The density of these point vortices is chosen in order that the flow remains tangent at midpoints between adjacent vortices and that the total vorticity around the obstacle is conserved.
In this talk, we discuss how to achieve a fully rigorous mathematical justification of this method for any smooth exterior domain. This is a joint work with Christophe Lacave and Emmanuel Dormy.
Hajer Bahouri - Dispersion phenomena on the Heisenberg group when the vertical frequency tends to 0
(from a joint work with Jean-Yves Chemin and Raphael Danchin)
In the first part of the talk, we build a frequency space on the Heisenberg and establish asymptotic description of the Fourier transform when the `vertical' frequency tends to 0. This construction is based on a new approach of the Fourier transform on the Heisenberg group which is classically defined as a one parameter family of bounded operators on $L^2(\R^d)$.
In the second part, we take advantage of this relevant approach to analyze dispersion phenomena on the Heisenberg group when the vertical frequency tends to 0.
Massimiliano Berti - Long time existence of space periodic water waves
In this talk we present the following result: any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size, is almost globally defined in time on Sobolev spaces, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. The proof is based on a paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders, combined with a subsequent normal form procedure. The reversibility structure of the water waves equations, and the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions. This is a joint work with J.M. Delort.
Tristan Buckmaster - A New Analytic Approach to Wave Turbulence
In this talk I will discuss an ongoing collaboration with Pierre Germain, Zaher Hani and Jalal Shatah regarding the large box limit for the non-linear Schrödinger on the torus.
Thierry Cazenave - Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value
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Yu Deng - Instability of 2D Couette flow in low regularity spaces
In 2013, Bedrossian and Masmoudi proved the stability and asymptotic stability of the Couette flow on T*R for small perturbations in Gevrey class G^(2-). Here, a natural questions is whether this high regularity requirement can be relaxed. In this talk, I will present recent work joint with Nader Masmoudi that establishes the optimality of this Gevrey exponent, by proving instability of the Couette flow for small perturbations in any Gevrey class G^(2+). The main ingredients in the proof are: a detailed study of the "toy model" introduced in the work of Bedrossian-Masmoudi, carefully selected perturbations and higher order perturbations, and a combination of physical space and Fourier space techniques.
Patrick Gérard - Low regularity solutions of the cubic Szegö equation
The cubic Szegö equation is a Hamiltonian PDE on the Hardy space displaying both integrability and instability properties, and
so far has been mainly studied on spaces of functions with high regularity. In this talk, I will show that, though it defines a flow
on the BMO Hardy space, it is illposed on the L^\infty Hardy space. I wll also discuss the problem of propagation of low Sobolev regularity.
This is a jointwork with Herbert Koch.
Tej-eddine Ghoul - On the harmonic map heat flow in supercritical dimensions.
We will give an overview of the recent results obtained on the harmonic map heat flow in supercritical dimensions.
Mahir Hadzic - Expanding large global solutions of the compressible Euler equations.
In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the free boundary compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time.
We show that if the adiabatic exponent gamma belongs to the interval (1, 5/3] then these affine motions are globally-in-time nonlinearly stable.
Our strategy relies on two key ingredients: a new interpretation of the affine motions using an (almost) invariant action of GL(3) on the compressible Euler system and the use of Lagrangian coordinates. The former suggests a particular rescaling of time and a change of variables that elucidates a stabilisation mechanism, while the latter requires new ideas with respect to the existing well-posedness theory for vacuum free boundary fluid equations. This is joint work with Juhi Jang (USC).
Hao Jia - Channels of energy for outgoing waves and asymptotics for energy
critical wave maps
The channel of energy type inequalities (for linear wave equations) introduced by Duyckaerts-Kenig-Merle have played a fundamental role in understanding the dynamics of energy critical wave equations. Most of those inequalities and applications are in the radial case, and many important questions remain open in the non-radial case. Recently a new
channel of energy inequality was found for outgoing waves in the non-radial case, which is quite robust. It seems to provide a new idea for the energy critical wave map equations. We will discuss this inequality and the
connection to wave maps. Joint work with Duyckaerts, Kenig and Merle.
Andrew Lawrie - Two-bubble dynamics for equivariant wave maps
We consider energy-critical wave maps taking values in the 2-sphere. It is known that initial data of topological degree zero and energy less than twice that of the ground state harmonic map leads to a global solution that scatters in both time directions. Under an equivariant symmetry reduction the same result holds for degree zero, k-equivariant data with energy less than twice that of the degree k harmonic map. In this talk we’ll consider data at the threshold energy. For any k-equivariant data with exactly twice the energy of the degree k harmonic map we prove that the solution is defined globally in time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps. The proof combines classical concentration-compactness techniques of Kenig-Merle with a modulation analysis of interactions between two harmonic maps in the absence of excess radiation. This is joint work with Jacek Jendrej.
Pierre-Louis Lions - On scalar conservation laws: old and new
Yasunori Maekawa - On the Prandtl boundary layer expansion in a Gevrey class
In this talk we discuss the verification of the Prandtl boundary layer expansion for the Navier-Stokes flows in the half plane under the noslip boundary condition. It is known that there is a generic instability mechanism in high frequencies
which prevents us verifying such an asymptotic expansion in the Sobolev class.
The goal of this talk is to establish the Prandtl expansion in a Gevrey class, around a monotone and convex shear boundary layer. This talk is based on a joint work with David Gerard-Varet and Nader Masmoudi.
Frank Merle - Inelasticity of soliton's collision for the energy critical wave equation
We prove that the collision of any two-solitons of the energy-critical wave equation always produces dispersion, for all possible parameters. This is a a joint work with Yvan Martel.
Clément Mouhot - Some applications of De Giorgi-Nash-Moser regularity theory in kinetic theory
Pierre Raphaël - Anisotropic blow up bubbles
I will consider the nonlinear heat equation in the energy super critical regime. I will explain how to construct a new class of blow up solutions which have a strong anisotropy in space: the rate of concentration dramatically depends on the considered direction in space. The proof requires revisiting the blow up analysis for both and type I and type II blow up bubbles.
Frédéric Rousset - Quasineutral limit for the Vlasov-Poisson system
We will study the Vlasov Poisson system for electrons or ions in the quasineutral regime.
This is a singular limit where many instabilities occur, in particular the limit system is not always well-posed. We will describe recent results obtained with D. Han-Kwan about the justification of this limit under some stability conditions.
Laure Saint-Raymond - Quasi-resonant forcing of internal waves in a domain with topography
(joint work with Yves Colin de Verdière)
Stratification of the density in an incompressible fluid is responsible for the propagation of internal waves.
In domains with topography, these waves exhibit interesting properties.
In particular, numerical and lab experiments show that in 2D these waves concentrate on attractors for some generic frequencies of the forcing (see Dauxois et al).
At the mathematical level, this behavior can be analyzed with tools from spectral theory and microlocal analysis.
Jalal Shatah - Invariant manifolds for quasilinear dispersive equations
Anne-Sophie de Suzzoni - Dispersion and the Dirac operator
In this talk, I will present some aspects of dispersion for the Dirac operator. I will start by partially reviewing what is known for the Dirac operator in a Minkowkski space-time. Then, I will introduce the Dirac operator in a curved space-time, and present a result of dispersion for specific cases such as assymptotically flat or warped product geometries.
This is a joint work with F. Cacciafesta (Padova).
Vladimir Sverak - On the Cauchy problem for vortex rings
We will discuss the existence and uniqueness for the Cauchy problem for the Navier-Stokes equation when the vorticity of the
initial datum is concentrated on a circle. (Joint work with Thierry Gallay.)
Isabelle Tristani - Perturbative Cauchy theory for the Landau and Boltzmann equations
In this talk, we will deal with both Landau and Boltzmann (without cutoff) equations on the torus. We develop a perturbative Cauchy theory around the equilibrium and treat the problem of convergence towards the equilibrium by studying linearised problems from a semigroup point of view. We will present the proofs of those existence theorems (the strategy is similar for both equations but not the tools used to follow it). This talk is based on joint works with Kleber Carrapatoso and Kung-Chien Wu; Frédéric Hérau and Daniela Tonon.
