The goal of the course is to introduce several tools to study the existence of solutions of nonlinear dispersive PDEs such as the Schr ̈odinger and wave equations. When these equations are on euclidean spaces, dispersion is often sufficiently strong to ensure the existence of global in time solutions with dispersive behaviour. On the contrary, on compact manifolds dispersion manifests in a weaker way and global in time results are harder to obtain. In the last part of the course we will see how probabilistic methods can help in globalizing solutions, and even construct local in time solutions when general results of local well posedness are not available. The tentative syllabus is the following:

1: Introduction

-Littlewood-Paley decomposition

-Dynamics of linear dispersive constant coefficients PDEs

2: Schr ̈odinger equation on euclidean spaces

- Strichartz estimates

- Local existence for subcritical and critical Schr ̈odinger equations

- Global existence results

- Introduction to blow-up solutions

3: Schr ̈odinger equation on compact manifolds

- Strichartz estimates with loss of derivatives

- Local well-posedness results: the Schr ̈odinger equation onTd

- Global well posedness reults in dimension 1 and 2

4: Probabilistic well-posedness

- Almost-sure global existence via invariant measures: Bourgain ’94

- Random data Cauchy theory: Burq-Tzvetkov ’08

References

[1] Bourgain, J.: Periodic nonlinear Schr ̈odinger equation and invariant measures.Comm. Math. Phys.166, 1–26, 1994.

[2] Burq, Tzvetkov: Random data Cauchy theory for supercritical wave equations I: local theory.Invent. math. 173, 449–475, 2008

[3] Burq, Gerard, Tzvetkov: Strichartz inequalities and the nonlinear Schrodinger equation on compact manifolds.American Journal of Mathematics, 126(3), 569–605, 2004

[4] Cazenave:Semilinear Schr ̈odinger Equations, Courant Lecture Notes in Mathematics 10, 2003.

[5] Erdogan, Tzirakis: Dispersive partial differential equations.London Mathematical society student text, 86, 2016.

[6] Linares, Ponce: Introduction to Nonlinear Dispersive Equations. Springer–Verlag, 2009

[7] Tao:Nonlinear Dispersive Equations. Local and Global Analysis. CBMS Regional Conference Series in Mathematics, 106, AMS (2006).

[8] Tzvetkov: Random data wave equations, arXiv:1704.01191