RG Analysis and Partial Differential Equations

V5B2 - Selected Topics in Analysis and PDE - Dispersive PDEs: deterministic and probabilistic perspectives

Summer Semester 2021

Dr. Leonardo Tolomeo
Instructor

Lectures

  • Tue 14-16 zoom
Every Tuesday there will be a lecture, taking place on Zoom. Recordings of the lectures, together with the handwritten notes taken during the lectures, will be made available on this website.


Topics

This course aims at providing the basis for the study of dispersive equations, both in the deterministic setting, and in the probabilistic one. Our goal is to show how probabilistic effects affect the behaviour of these equations, greatly improving the results available. We will cover

1. Strichartz estimates for Schrödinger and wave equations.
2. Local well posedness theory Schrödinger and wave equations in subcritical Sobolev spaces H^s.
3. Global well posedness theory for Schrödinger and wave equations in H^1.
4. Ill posedness in supercritical Sobolev spaces.
5. Local well posedness in supercritical Sobolev spaces for random initial data.
6. Global well posedness for random initial data.

If time allows, we will also discuss some features of the associated stochastic PDEs.


Prerequisites

Analysis: basic real and complex analysis, basic knowledge of Fourier analysis.
Probability: measure theoretical approach to probability, Gaussian random variables, independence.

Literature

  • L. Grafakos, Classical and modern Fourier analysis.
  • T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis.
  • M. Gubinelli, T. Souganidis, N. Tzvetkov, Singular Random Dynamics, Chapter 4.