Escuela de Ecuaciones Diferenciales Parciales Hamiltonianas

Abstract: I will present a new method based on normal form and psudodifferential calculus to get spectral results on Schrödinger type operators on $T^d$. In the one dimensional case one obtains in a very simple way the classical result that the eigenvalues of a Schroedinger operator come in couple which are well separated one from the others and such that the two eigenvalues in a couple have the same asymptotic.

In the higher dimensional case I will show how to obtain the asymptotic behavior of a large part of the spectrum of Schroedinger operators and prove some properties similar to those just described for the one dimensional case.

Abstract: In this mini course I will present different normal forms technics and their applications to the dynamics of nonlinear partial differential equations (PDE).

In the first lecture I will emphasize on the Birkhoff normal form (BNF) in finite dimension giving a (almost) complete proof.

Then in the second lecture I will present some applications in the PDE context.

During the third lecture I will propose a variant of the BNF which allows to control the low Fourier modes of the small solutions of the nonlinear wave equation on the d dimensional torus (d\geq 2). This last technique can also be applied to certain Boussinesq systems that appear in the theory of water waves.

Abstract: In these lectures we will focus on some singular dynamics for incompressible inviscid fluids, called vortex dynamics. In the 2D case, such solutions correspond to a vorticity behaving as a sum of Dirac masses centered at points called point vortices. In the 3D case, these solutions correspond to a vorticity concentrated on curves called vortex filaments. In the 2D case, we will present some mathematical results asserting that the point vortices evolve according to an Hamiltonian system of ordinary differential equations (point vortex system), for which we will study the mathematical properties. We will then introduce the binormal curvature flow equation, which formally governs the motion of one single vortex filament. Finally, we will present a system of simplified equations derived by Klein, Majda and Damodaran to modelize the interaction of several almost parallel vortex filaments.

Abstract: We will discuss some major mathematical problems in the theory of statistical mechanics of nonlinear dispersive waves, commonly known as wave turbulence. Such questions range from deriving and justifying the fundamental kinetic equations of wave turbulence, all the way to understanding the conclusions of this theory like energy cascade phenomena. We will try to give a survey of the recent developments on such fronts.