V5B1 - Advanced Topics in Analysis and PDEs - Transport Equations and Fluid Dynamics
Summer term 2024/2025
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.
Overview
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.
IMPORTANT NOTE: there will be no lecture on the first week of April (01.04.2025 and 03.04.2025). Lectures will start on Tuesday 11.04.2025 at 10h.
News
Daniel Huybrechts elected as member of Leopoldina
Catharina Stroppel appointed Honorary Doctor at Uppsala University
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