Introduction to Mathematical Fluid Dynamics

In this lecture, we will be concerned with the mathematical theory for the Euler and Navier-Stokes equations. It has two main topics.

In the first part of the lecture, we will study the (local) existence and uniqueness of weak solutions and discuss possible accumulation of vorticity.

In the second part, we concentrate on the Euler equations in two spatial dimensions and investigate the motion of vorticity fields. We show the stability of so-called Lamb-Oseen dipoles and infer a system of ordinary differential equations from the microscopic equations that describe the interaction of point vortices.

Conceptually, we will need a number of mathematical tools, that will be developed during the lecture. These include

• Compactness and weak convergence in Lp spaces
• Rearrangement inequalities
• Lp regularity estimates

This lecture is a continuation of a standard introductory course in partial differential equations.

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: seis@wwu.de

Topics in Mathematical Fluid Dynamics

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

Depending on the background of the participants, we will discuss basic concepts or feasible research papers in the mathematical theory of fluid dynamics. Participating students should be familiar with partial differential equations on the level of a standard introductory course (e.g., PDGL 1).

The Seminar can lead to a Master's thesis.

Interested students should contact us by email seis@wwu.de

Advanced Topics in ANALYSIS & PDEs

This is a reading seminar that we run together with  André Schlichting's group. It runs all year round. We read mathematical papers related to mixing and turbulence, transport and diffusion, or whatever looks interesting to (at least) one of us. Some papers are classics, some are brand new on the arXiv.

We meet currently on Fridays at 10:00. Interested? Then write us an email!

Update November 2022: We return to the deterministic point of view and study hypocoercivity based on Villani's monograph. This will take some time...

Here is an overview of past topics:

• 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