Nonlinear partial differential equations

Tuesdays and Fridays 8-10 in lecture hall M5.

Are you curious about the mathematics behind fluid motion? This course dives into the Euler equations, the cornerstone of inviscid fluid dynamics, and explores both their rigorous analysis and fascinating phenomena.

We will cover:

• Well-posedness – when do solutions exist, and when are they unique?
• Blow-up criteria – what mechanisms can lead to singularity formation?
• Stability of steady flows – how robust are equilibria under perturbations?
• Vortex sheets – sharp interfaces and their instabilities.
• Point vortex models – can we predict the path of a tornado?

The course balances theory and intuition, making it ideal for students interested in analysis, PDEs, and mathematical physics.

Prerequisites for this course include fundamental knowledge on PDEs including Sobolev theory.

Feel free to contact me if you need any further information.

Partielle Differentialgleichungen und ihre Anwendungen

In diesem Bachelor-Seminar beschäftigen wir uns mit elementaren Problemen im Bereich partieller Differentialgleichungen, sowie ersten Anwendungen der mathematischen Theorie auf Probleme der Physik. Grundlegende Kenntnisse von Partiellen Differentialgleichungen sind erwünscht; diese können aber auch parallel zum Seminar erworben werden.

Das Seminar kann auf eine Bachelorarbeit in diesem Gebiet vorbereiten.

Bei Interesse, kontaktieren Sie uns gerne via Email.

Topics in Mathematical Fluid Dynamics

This Master’s seminar introduces mathematical fluid dynamics through the lens of vortex dynamics for the Euler equations.

We will read and discuss recent research papers that are chosen to be accessible and enjoyable, offering a first step into the world of mathematical research. The focus is not only on the results, but also on learning how to approach and understand modern articles in analysis, PDEs, and fluid dynamics.

The seminar is an opportunity to get a taste of current developments, build confidence in reading mathematical literature, and possibly prepare the ground for a Master’s thesis project.

Interested students are encouraged to contact me by email for further details.

Advanced Topics in ANALYSIS & PDEs

This is our reading seminar, currently organized by Björn Gebhard.

The seminar runs year-round, and we read mathematical papers on topics such as mixing and turbulence, transport and diffusion, or anything that catches the interest of at least one of us. Some papers are classics, while others are brand new from the arXiv.

This autumn we will start a new focus on paradifferential calculus.

We currently meet on Tuesdays at 10. Interested? Send us an email!

Below is an overview of past topics:

• Spring 2025: Isoperimetric problems with Coulomb interaction
• 2024: A full year on vortex theory in fluid dynamics
• Winter 2023: Our new focus is on Landau damping.
• March 2023: Back to qualitative properties: Our next topic is Hörmander's hypoellipticity.
• November 2022: We return to the deterministic point of view and study hypocoercivity based on Villani's monograph. This will take some time...
• January 2022: We start the new year with a focus on Random Dynamical Systems by reading Arnold's book, studying Harris' theorem and discussing recent papers on mixing by stochastically driven flows.
• Nov 22, Dec 6 & 13: Víctor talks about the well-posedness of the 2D Navier-Stokes equations with measure-valued initial data
• Oct 25, Nov 8 & 15: Dominik describes an alternative approach to sharp extinction rates
• Oct 8, 11 & 18: Christian presents a new work on sharp extinction rates for fast diffusions on bounded domains
• June 30: Víctor presents his new work on stability estimates for advection-diffusion equations in the DiPerna-Lions setting and beyond
• May 31, June 9 and 16: André speaks about the mean-field limit for Coulomb-type flows
• April 30 & May 12: Dominik presents results on the large-time asymptotics for the thin film equation, its gradient flow interpretation and its relation to the porous medium equation
• March 26, April 1 & 21: Stefano presents Schochet's paper on the Euler mean field limit
• Feb 12, March 5, 12 & 19 : Christian gives an introduction on vortex sheets and reads the classical Delort paper on measure valued solutions to Euler
• Jan 22, 29 & Feb 5-- HAPPY NEW YEAR 2021!! -- We resume our seminar with a talk on universal mixers. Víctor's speaking!
• Dec 11 & 18: Dominik presents a paper on invariant manifolds for Navier-Stokes
• Nov 20, 27 & Dec 4: Stefano presents his results on the vortex dynamics for viscous fluids
• Oct 30, Nov 6 & 13: Christian speaks about large time asymptotic expansions for the porous medium equation and the role of the spectrum
• Oct 16 & 23: Dominik talks about large time asymptotics for the porous medium equation via the entropy method
• Oct 2 & 9: Víctor talks about integrability estimates for advection-diffusion equations and their optimality