Partial differential equations

In this lecture, we will explore several fundamental partial differential equations, including:

• linear transport and continuity equations,
• Burgers' equation,
• Laplace's and Poisson's equations,
• the heat equation, and
• the wave equation.

We will derive key properties such as maximum and comparison principles, maximal regularity, smoothing and propagation of regularity, as well as blow-up and shock formations, and equilibration.

Various techniques for finding solutions will be introduced, including variational methods, fixed-point methods, and Green’s functions.

Prerequisites for this course include basic tools from functional analysis, particularly Sobolev theory.

I can provide materials for a self-study crash course on Sobolev theory, which can be reviewed either before or alongside this lecture as a refresher.

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 graduate seminar is designed for students in the Master's programs in Mathematics or Physics.

This term, we will explore the De Giorgi-Nash-Moser theory, a powerful framework in the regularity theory of partial differential equations. This theory addresses the Hölder continuity of weak solutions to elliptic and parabolic equations under minimal regularity conditions on the data. It was instrumental in solving Hilbert’s nineteenth problem and has since spurred numerous significant developments in mathematics. Our focus will be on the elliptic case, where we will examine the distinct approaches developed by De Giorgi, Nash, and Moser.

The seminar can also serve as a foundation for a Master's thesis.

Interested students are encouraged to contact Lukas Niebel via email.

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 fall, we're resuming after a brief break, with our new focus on vortex theory in fluid dynamics.

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

Below is an overview of past topics:

• 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: Domink 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