Séminaire EDP-Analyse ICJ

Existence results for the higher order Q-curvature equation / Probabilistic local well-posedness for nonlinear Schrödinger equations.

par Louise Gassot (CNRS, IRMAR, Rennes), Saikat Mazumdar (IIT Bombay)

Europe/Paris
Fokko (ICJ)

Fokko

ICJ

Description
  1. In this talk, we will obtain some existence results for the Q-curvature equation of arbitrary 2k-th order, where k ≥ 1 is an integer, on a compact Riemannian manifold of dimension n ≥ 2k + 1. This amounts to solving a nonlinear elliptic PDE involving the powers of Laplacian called the GJMS operator. The difficulty in determining the explicit form of this GJMS operator together with a lack of maximum principle complicates the issues of existence. This is a joint work with Jérôme Vétois (McGill University).
  2. We discuss ill-posedness in supercritical regimes for the nonlinear Schrödinger equation, that manifests as a norm inflation mechanism. We compare this phenomenon to probabilistic well-posedness for random initial data in these regimes. Finally, we establish almost-sure local well-posedness below the energy space for a two-dimensional anisotropic Schrödinger Half-wave equation with cubic nonlinearity. Because of the lack of probabilistic smoothing in the first Picard’s iterate due to the high-low-low nonlinear interactions, we use a refined probabilistic ansatz initiated by a method from Bringmann. This is a joint work with Nicolas Camps and Slim Ibrahim.