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 cover1. 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.
News
Opening Hausdorff Chair with deadline March 15, 2025
Angkana Rüland receives Gottfried Wilhelm Leibniz Prize 2025
Wolfgang Lück receives the von Staudt Prize
Gerd Faltings elected member of the Order Pour le Mérite
Geordie Williamson receives the Max Planck-Humboldt Research Award 2024
ERC Starting Grant for Markus Hausmann
EMS Prize 2024 for Jessica Fintzen
Bonn mathematics performs excellently again in QS ranking
Stefan Schwede is invited speaker at the ECM 2024 in Sevilla
Jessica Fintzen wins Cole Prize
Catharina Stroppel receives Gottfried Wilhelm Leibniz Prize 2023