Mathematical Institute of the University of Bonn

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 on zoom

Every Tuesday there will be a lecture, taking place on Zoom. The handwritten notes taken during the lectures will be made available on this website. Recordings of the lectures are available, please contact Dr. Tolomeo via email if you are interested. For convenience, the recording of lecture 1 (and only lecture 1) is available on this page.

• Notes of lecture 1, Recording of Lecture 1,
• Notes of lecture 2,
• Notes of lecture 3,
• Notes of lecture 4,
• Notes of lecture 5,
• Notes of lecture 6,
• Notes of lecture 7,
• Notes of lecture 8,
• Notes of lecture 9,
• Notes of lecture 10,
• Notes of lecture 11,
• Notes of lecture 12,
• Notes of lecture 13.

The information about the zoom session can be found on Basis.

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.