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 Nicolas Camps and Charles Collot.
Invariance of the Gibbs measure for NLS on the sphere
by Nicolas Camps
These are ongoing joint works with Nicolas Burq, Chenmin Sun and Nikolay Tzvetkov.
The two talks are devoted to a statistical approach to nonlinear waves on surfaces. Specifically, we consider the cubic Schrödinger equation (NLS) on the 2 dimensional sphere and we study the collective dynamics of initial data distributed according to the formally invariant Gibbs measure.
In the first presentation, we provide a brief overview of the Cauchy problem for NLS posed on compact surfaces. Then, we introduce the Gibbs measure problem and a general resolution scheme developed by J. Bourgain in 1996, in the particular case of the torus. In the case of the sphere, however, we uncover strong instabilities due to concentrated spherical harmonics around a great circle, preventing a direct extension of Bourgain’s method to the sphere.
The second talk introduces a probabilistic quasi-linear resolution scheme with a critical flavor, which is tuned to overcome the instabilities and to solve the Gibbs measure problem for the sphere.
On type II singularity formation for the parabolic-elliptic Keller-Segel system
by Charles Collot
The parabolic-elliptic Keller-Segel system is a semilinear advection-reaction-diffusion equation, introduced by Patlak, Keller and Segel, that models the evolution of bacteria density under chemotaxis. It is one of the most studied models for singularity formation among parabolic equations. It preserves the mass of the solutions, has a scaling invariance, and is critical in 2 dimensions and supercritical in higher ones. It admits type I and type II singular solutions, which refers to different blow-up rates near the singular time.
The aim of the mini-course is to present novel techniques for showing the existence of type II singular solutions, and for studying their stability. They are based on joint works with T.-E. Ghoul (NYU Abu Dhabi), N. Masmoudi (NYU Abu Dhabi) and V. T. Nguyen (National Taiwan University), and on an ongoing work with them. It will be divided in 4 parts.
In part 1), we will define type I and type II singular solutions. We will review the known results on the existence of such solutions, both in critical and supercritical dimensions. On these examples we will see a difference in the type of self-similarity that the solution displays close to the singularity.
In part 2), we will explain how to construct an approximate type II blow-up solution in two dimensions. This will use matched asymptotics, akin to the tail dynamics computations in the works of Raphael-Schweyer, Merle-Raphael, Raphael-Rodnianski etc. We will relate this construction to the spectral analysis of the linearised operator around a concentrating ground state in the parabolic neighbourhood of the singularity.
In part 3), we will study the stability of the approximate solution. We will present a novel idea of “matched scalar product”, which is a functional spaces’ analogue of matched asymptotics. At the ground state’s scale, it ressembles the quadratic form associated to the linearization of the free energy functional (for which the equation is a gradient flow for the Wasserstein metric), and at the parabolic scale it ressembles the Gaussian L2 space associated to a Fokker-Planck operator.
In part 4), if time permits, we will discuss the construction of a singular solution with colliding bubbles. This dynamics was predicted at a formal level by Velazquez-Seki-Sugiyama.
The Hele-Shaw semi-flow
Abstract : This talk is about a joint work with Herbert Koch. We prove that the Cauchy problem for the Hele-Shaw equation is well-posed in a strong sense and in a general setting. Our main result is the construction of an abstract semi-flow for the Hele-Shaw problem within general fluid domains (enabling, for instance, changes in the topology of the fluid domain) and which satisfies several properties: comparison principle, stability estimate, monotonicity and continuity results, along with many Lyapunov functionals. We establish numerous qualitative properties, eventual analytic regularity result for any arbitrary initial data. When the domain is the subgraph of a function, we prove a global regularity result for initial data in sub-critical Sobolev spaces, well-posedness in a strong sense for initial data with barely a modulus of continuity, as well as waiting-time phenomena for Lipschitz solutions, in any dimension.
Almost global existence for some nonlinear Hamiltonian PDEs on some higher dimensional manifolds
Abstract : Consider a manifold in which the geodesic flow is integrable and admits global action variables. I will present a result of almost global existence for small and smooth solutions of some semi-linear PDEs in such manifolds. Examples of manifolds fulfilling the assumption are flat tori, Lie groups and their homogeneous spaces, rotation invariant surfaces. The abstract result will be applied to the NLS, the Beam equation and to the problem of Hs stability of the ground state in NLS. This is joint work with Roberto Feola, Beatrice Langella and Francisco Monzani.
Quadratic operators, splitting methods and smoothing effects
Abstract : We consider linear evolution equations composed of a diffusive/ regularizing (degenerate) part and of a transport/ dispersive part. Focusing on the case of the non-self adjoint quadratic differential operators, I will explain how and why the interactions between these two dynamics enhance the regularizing effects of the equation. In particular, I will explain why, in this context, splitting methods are well suited to analyze this type of phenomena. This talk will rely on some joint works with Paul Alphonse.
Unstable Stokes waves
Abstract : In this talk I will describe results about modulational instability of Stokes waves.
Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field
Abstract: We consider a modulated magnetic field, B(t) = B_0 + epsilon f(omega t), perpendicular to a fixed plane, where B_0 is constant, epsilon>0 and f a periodic function on the n-dimensional torus.
Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for “almost all” non resonant frequencies omega. On the contrary, for the symmetric gauge we obtain that, for “almost all” non resonant frequencies omega, the Landau Hamiltonian is reducible to a two dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.
Joint work with: D. Bambusi, A. Maspero, D. Robert and C. Villegas-Blas
Exotic rotation domains and Herman rings for quadratic Hénon maps
Abstract: Quadratic Hénon maps are polynomial automorphism of C2 of the form h:(x,y)–> (lambda1/2(x2+c)-\lambda y,x). They have constant Jacobian equal to lambda and they admit two fixed points. If \lambda is on the unit circle (one says the map h is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel Theorem shows (under a Diophantine assumption) that h admits many quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, Ushiki observed some years ago what seems to be quasi-periodic orbits (though no Siegel disks exist). I will explain why this is the case. This theoretical framework also predicts (and proves), in the dissipative case (\lambda of module less than 1), the existence of (attractive) Herman rings. These Herman rings, which were not observed before, can be produced in numerical experiments.
Strong ill-posedness in L\infty of the 2D Boussinesq equations
Abstract: In this talk I will present a recent work in which the strong ill-posedness of the two-dimensional Boussinesq system is proven. I will show explicit examples of initial data with vorticity and density gradient in L\infty(R2) for which the horizontal density gradient has a strong norm inflation in infinitesimal time. This is a joint work with Roberta Bianchini (CNR) and Lars Eric Hientzsch (Bielefeld University).
The nonlinear shallow water equations with a partially immersed obstacle
Abstract : This talk (based on an joint work with T. Iguchi) is devoted to the proof of the well-posedness of a model describing waves propagating in shallow water in horizontal dimension d=2 and in the presence of a fixed partially immersed object. We first show that this wave-interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain, with boundary conditions that are fully nonlinear and nonlocal in space and time. This hyperbolic initial boundary value problem is characteristic, does not satisfy the constant rank assumption on the boundary matrix, and the boundary conditions do not satisfy any standard form of dissipativity. Our main result is the well-posedness of this system for irrotational data and at the quasilinear regularity threshold. In order to prove this, we introduce a new notion of weak dissipativity, that holds only after integration in time and space. This weak dissipativity allows high order energy estimates without derivative loss; the analysis is carried out for a class of linear non-characteristic hyperbolic systems, as well as for a class of characteristic systems that satisfy an algebraic structural property that allows us to define a generalized vorticity. We then show, using a change of unknowns, that it is possible to transform the linearized wave-interaction problem into a non-characteristic system, which satisfies this structural property and for which the boundary conditions are weakly dissipative. We can therefore use our general analysis to derive linear, and then nonlinear, a priori energy estimates. Existence for the linearized problem is obtained by a regularization procedure that makes the problem non-characteristic and strictly dissipative, and by the approximation of the data by more regular data satisfying higher order compatibility conditions for the regularized problem. Due to the fully nonlinear nature of the boundary conditions, it is also necessary to implement a quasilinearization procedure. Finally, we have to lower the standard requirements on the regularity of the coefficients of the operator in the linear estimates to be able to reach the quasilinear regularity threshold in the nonlinear well-posedness result.
A scattering operator for some nonlinear elliptic equations
We consider non linear elliptic equations of the form \Delta u = f(u,\nabla u) for suitable analytic nonlinearity f, in the vinicity of infinity in Rd, that is on the complement of a compact set.
We show that there is a one-to-one correspondence between the non linear solution u defined there, and the linear solution uL to the Laplace equation, such that, in an adequate space, u – uL_> 0 as |x|->+\infty. This is a kind of scattering operator.
On asymptotic stability for self-similar blowup of mass supercritical NLS
Abstract : For slightly mass supercritical nonlinear Schrodinger equations (NLS), self-similar blowup has been proven to exist and generate stable blowup dynamics. Towards the asymptotic stability, I first showed a finite codimensional version by introducing Strichartz estimate for the linearized matrix operator; in a forthcoming work, I count all the unstable directions of the matrix operator and then prove the asymptotic stability without losing codimensions. New techniques are introduced to determine the spectrum for such non-self-adjoint and non-compactly perturbed operator, which is expected to be useful also in other context.
Some examples of non unique minimizing travelling waves
Abstract : We consider nonlinear Schrödinger equations with a nontrivial condition at infinity for a large class of nonlinearity. It has been shown by Mihai Maris that these equations admit travelling wave solutions for any subsonic speed, and they are constructed as the solutions of a minimizing problem. We are interested in the following question : do we always have uniqueness of the solution of this minimizing problem, up to the natural invariances of the problem ? In this talk, we will show how to construct a specific nonlinearity for which this is not the case. This is a current project in collaboration with Mihai Maris.
Ayman Rimah Said
Small scale creation of the Lagrangian flow in 2d perfect fluids
Abstract: In this talk I will present a recent result showing that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time at least like t1/3. This initial data dependent norm quantifies the exact L2 decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade whichI I will then show to be the quantitative phenomenon behind a microlocal generalised Lyapunov function constructed by Shnirelman.
The binormal flow and the desingularization of the Biot-Savart integral
Abstract: I’ll present the so called Localized Induction Approximation that describes the dynamics of a vortex filament according to the Binormal Curvature Flow (BF). I’ll give a result about the desingularization of the Biot-Savart integral proved with Marco A. Fontelos within the framework of Navier-Stokes equations. Some particular examples regarding BF obtained with Valeria Banica will be also considered.