Seminar Analysis and Mathematical Physics: Nicolas Camps (University of Nantes)

Some probabilistic approaches for NLS in the Euclidean space

Following the seminal work of Bourgain in 1996, and Burq and Tzvetkov in 2008, a statistical approach to nonlinear dispersive equations has developed in various contexts. We are interested here in Schrödinger equations with cubic nonlinearity (NLS) in R^d. We first recall the relevant probabilistic Cauchy theory  developed by Bényi, Oh and Pocovnicu in 2015 in supercritical regimes, before specifying the norm inflation instability that occurs in this context. The second part is dedicated to long-time dynamics for solutions initiated from these randomized initial data. We demonstrate a scattering result that relies on a probabilistic version of the I-method and that allows to solve statistically the scattering conjecture for NLS in dimension 3. Finally, we present recent developments in quasi-linear regimes, which were initiated by Bringmann in 2019 and which we exploit to exhibit strong solutions to some weakly dispersive equations. This last result is in collaboration with Louise Gassot and Slim Ibrahim.

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