Welcome of participants Room: Cafeteria (ground floor)

Scattering for nonlinear Vlasov equations I

• Daniel Han Kwan (Université de Nantes)

Room: Room 16 (ground floor) The minicourse will be about the long-time behaviour of solutions to the Vlasov--Poisson or Vlasov--screened Poisson equations, set in the whole space. We will mostly try to review: (1) global existence and (modified) scattering for small initial data; (2) scattering in the screened case for initial data close to stable homogeneous equilibria. The approach we will follow for both problems is based on the study of the characteristics associated with Vlasov equations.

On the asymptotic analysis of the Einstein-Vlasov system

• Jacques Smulevici

Room: Room 16 (ground floor) The Einstein-Vlasov system is the natural generalization of the classical Vlasov-Poisson system in the context of the general theory of relativity. I will start by an introduction to the Einstein equations and relativistic (collisionless) kinetic theory. I will then present several results concerning the asymptotics and the global existence of small data solutions for the Einstein-Vlasov system with either massive or massless particules.

Scattering for nonlinear Vlasov equations II

• Daniel Han Kwan

Room: Room 16 (ground floor) The minicourse will be about the long-time behaviour of solutions to the Vlasov--Poisson or Vlasov--screened Poisson equations, set in the whole space. We will mostly try to review: (1) global existence and (modified) scattering for small initial data; (2) scattering in the screened case for initial data close to stable homogeneous equilibria. The approach we will follow for both problems is based on the study of the characteristics associated with Vlasov equations.

Scattering theory for NLS I

• Nicola Visciglia

Room: Room 16 (ground floor) We study the long time behaviour of solutions to NLS. We shall first show that for long-range nonlinearities the nonlinear solutions cannot behave as free waves for large times. Next we shall focus on the short range case and we shall describe several results proving that nonlinear solutions behave as free wave in a suitable sense. In particular we shall discuss the scattering theory in H^1 and in \Sigma spaces. We shall also present during the lectures some open problems.

Stability results for resonant Schrödinger equations on Diophantine tori

• Nicolas Camps

Room: Room 16 (ground floor) This presentation is devoted to a stability result for cubic Schrödinger equations (NLS) on Diophantine tori. We prove that the majority of small solutions in high regularity Sobolev spaces do not exchange energy from low to high frequencies over very long time scales. We first provide context on the Birkhoff normal form approach in the study of the long-time dynamics of the solutions to Hamiltonian partial differential equations. Then, we present the induction on scales normal form which is at the heart of the proof. Throughout the iteration, we ensure appropriate non-resonance properties while modulating the frequencies (of the linearized system) with the amplitude of the Fourier coefficients of the initial data. Our main challenge is then to addressing very small divisor problems, and describing the set of admissible initial data. The results are based on a joint work with Joackim Bernier, and an ongoing joint work with Gigliola Staffilani.

Scattering theory for NLS II

• Nicola Visciglia

Room: Amphi Lebesgue (ground floor) We study the long time behaviour of solutions to NLS. We shall first show that for long-range nonlinearities the nonlinear solutions cannot behave as free waves for large times. Next we shall focus on the short range case and we shall describe several results proving that nonlinear solutions behave as free wave in a suitable sense. In particular we shall discuss the scattering theory in H^1 and in \Sigma spaces. We shall also present during the lectures some open problems.

Large-time behavior for the Landau equation

• Kléber Carrapatoso

Room: Amphi Lebesgue (ground floor) I will present recent results for the Landau equation that prove the converge of solutions to the unique Maxwellian equilibrium (with quantitative rates).

Stability Analysis of Quantum Dissipative Systems

• Simona Rota Nodari

Room: Amphi Lebesgue (ground floor) In this talk I will present the stability properties of plane wave solutions for a system describing quantum particles interacting with a complex environment. From a mathematical point of view, this amounts to studying a system of PDEs coupled in a non-local (in time and space) way, which complicates the analysis considerably compared to the usual nonlinear Schrödinger equations. The strategy adopted is based on the identification of suitable Hamiltonian structures and Lyapunov functionals. Work in collaboration with T. Goudon.

Focusing dynamics of 2D Bose gases in the instability regime

• Charlotte Dietze

Room: Amphi Lebesgue (ground floor) We consider the dynamics of a 2D Bose gas with an interaction potential of the form $N^{2\beta-1}w(N^\beta\cdot)$ for $\beta\in (0,3/2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schrödinger equation (NLS) may blow up in finite time. We show that to leading order, the $N$-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics. This is joint work with Lea Boßmann and Phan Thành Nam.

Semiclassical analysis of quantum scattering in the Yukawa interaction

• Zied Ammari

Room: Room 16 (ground floor) In this talk, I will consider the quantum scattering theory of a Yukawa particle field model in the semi-classical regime. The transition and diffusion amplitudes of such a model will be linked to those of the Schrödinger--Klein--Gordon equation. The phase space properties of asymptotic vacuum states will be highlighted and the concentration around the radiation-free solutions of the SKG equation will be established. I will also discuss the uniqueness of SKG energy minimizers as well as some open questions related to asymptotic completeness. This is a joint work with Marco Falconi and Marco Olivieri.

Free time (PDE seminar) / coffee break Room: Amphi Lebesgue (ground floor)

Scattering theory of classical particles

• Jan Derenzinski

Room: Room 16 (ground floor) I will speak about classical scattering theory, both 2-body and N-body. It is much less developed than its quantum counterpart. Nevertheless, it has some interesting constructions and results, which are later useful in the quantum theory. Therefore, apart from its independent interest, my talk can serve as an introduction to Christian Gerard's lectures on quantum N-body scattering.

Scattering for the quantum $N$-body problem I

• Christian Gérard

Room: Room 16 (ground floor) We will review the scattering theory for the quantum $N$-body problem, focusing on asymptotic completeness. We will explain the methods developped in the eighties and nineties to tackle this problem, which led to a complete solution of the asymptotic completeness problem.

A many-body RAGE theorem

• Jonas Lampart

Room: Room 16 (ground floor) The RAGE theorem is a general statement on the relation of spectrum and dynamics for self-adjoint operators. It states that the large-time dynamics converge, in a certain weak sense, to the projection of the initial condition onto the bound states of the generator. The contributions of the continuous spectrum disappear in this limit. The many-body version of the theorem, which is joint work with Mathieu Lewin, refines this statement in the case of N-body systems. In an appropriate sense, these systems converge for large times to a combination of bound states of systems with n=0,..,N particles. Including the decomposition into subsystems with strictly fewer particles resolves the contribution of the continuous spectrum on the N body generator, which is due to freely moving clusters of particles.

Scattering for the quantum $N$-body problem II

• Christian Gérard

Room: Amphi Lebesgue (ground floor) We will review the scattering theory for the quantum $N$-body problem, focusing on asymptotic completeness. We will explain the methods developped in the eighties and nineties to tackle this problem, which led to a complete solution of the asymptotic completeness problem.

Some dynamical properties of the scattering operator for NLS

• Rémi Carles

Room: Amphi Lebesgue (ground floor) For the nonlinear Schrodinger equation with short range nonlinearity, the construction of this operator often implies sone analyticity properties of this operator, as we will explain in a first part. We will also show how, in the special case of the mass-critical nonlinearity, it is rather straightforward to construct final states, or initial data, for which the action of the scattering operator, or inverted wave operator, is explicit, consisting in the multiplication by an arbitrary complex number of modulus one.