Mathematical Institute of the University of Bonn

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

Summer term 2024/2025

Dr. Ruoyuan Liu

Organisational details

• Time and place: Friday 10h-12h, SemR 1.008

• If there are any questions, please get in touch with the lecturer. If you are a participating Masters student, please register on Basis.

  Basis page

Overview

The lecture aims to provide the basics of dispersive partial differential equations (PDEs) from both an analytic point of view and also a probabilistic point of view. Dispersive PDEs are a class of PDEs where different frequencies propagate at different velocities. We shall mainly use Schrödinger equations and wave equations as examples and use tools from harmonic analysis and probability theory to study well-posedness (existence, uniqueness, and stability under perturbation) of these equations.

The lecture is divided into two main parts. In the first part of the lecture, we discuss deterministic well-posedness theory of nonlinear Schrödinger equations and nonlinear wave equations. A highlight of this part is the Strichartz estimates which allow us to show well-posedness in (sub-)critical regimes. In the second part, we discuss probabilistic well-posedness theory of nonlinear Schrödinger equations and nonlinear wave equations using the randomized initial data, which goes beyond deterministic well-posedness results.

Practical Details

The syllabus contains additional details, including a list of prerequisites, a tentative schedule and literature recommendations.