Mathematical Institute of the University of Bonn

V5B1 - Advanced Topics in Analysis and PDEs - Transport Equations and Fluid Dynamics

Summer term 2024/2025

Dr. Dimitri Cobb

Organisational details

• Time: Tuesday 10h-12h and Thursday 12h-14h.

• 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

Syllabus

The unifying topic of this course is the transport equation: one of the simplest linear Partial Differential Equations (PDEs), which is in a way the PDE analogue of ODEs. Despite being extremely simple and linear, the transport equation is ubiquitous in many models from mathematical physics, especially fluid dynamics, as it represents the evolution of a set of particles in a (fluid) flow. As such, its properties have been intensively studied throughout the past decades, are still the topic of active research, and the results concerning it are routinely used in the analysis of non-linear PDEs.

The lectures will reflect the two sides of this story, as we will alternate between studying different aspects of the theory of the linear transport equation, and pairing each of these with an application to the non-linear PDEs of fluid dynamics. This will mainly involve examining how the solution is affected by the regularity of the vector field (if it is smooth, Lipschitz, Sobolev, log-Lipschitz, etc.). In doing so, we will introduce many of the important tools of harmonic analysis that are used in the modern study of PDEs (Singular Integral Operators, commutators, Besov Spaces and Littlewood-Paley analysis, etc.).

Practical Details

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

Here are some handwritten lecture notes. These contain many mistakes, and are not necessarily a faithful representation of what is taught in the lecture (especially concerning the order of the material), and therefore cannot replace notes taken in class. Use at your own risk...

Here is a crash course on weak topology and distributions, which also serves as a reference for functional analysis theorems we will be using.

IMPORTANT NOTE: there will be no lecture on 01.05.2025 and 06.05.2025.