Nikolay Tzvetkov

Ill-Posedness Issues for the Benjamin--Ono and Related Equations

We establish that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula. As a consequence, the flow map fails to be smooth.

Random data Cauchy theory for supercritical wave equations II: a global existence result

We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in

Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces

bigcap_{s<1/2} (H^s(Theta)times H^{s-1}(Theta))

Random data Cauchy theory for supercritical wave equations I: local theory

. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations, inspired by earlier works by Bourgain [2, 3] on the non linear Schrödinger equation, which allow us to obtain also precise large time dynamical informations on our solutions.

Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution

We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere

Lp properties for Gaussian random series

mathbb{S}^2

Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations

and more generally on Zoll surfaces, we prove that, for s>1/4, NLS is uniformly well-posed in Hs, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. RésuméOn étudie l’équation de Schrödinger non linéaire (NLS) sur une surface compacte. Sur la sphère

On nonlinear Schrödinger equations in exterior domains

mathbb{S}^2

Transverse nonlinear instability for two-dimensional dispersive models

et plus généralement sur toute surface de Zoll, on démontre que pour s>1/4, NLS est uniformément bien posée dans Hs, ce qui est optimal sur la sphère. Le principal ingrédient de notre démonstration est une estimation bilinéaire pour les projecteurs spectraux du laplacien sur une surface compacte.

Invariant measures for the nonlinear Schrödinger equation on the disc

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in Hs(M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in Hs(M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary.