Research group

Analysis and PDEs

Prof. Dr. Christian Seis

Office: Orléans-Ring 10, 48149 Münster

Mail addess: Einsteinstr. 62, 48149 Münster

What's going on??

Our paper on invariant manifolds for the thin film equation was published in the journal Archive for Rational Mechanics and Analysis. That's cool!

In a joint work with Nico de Nitti (Lausanne) and David Meyer, we investigated the optimal regularity that can be propagated by bounded vorticity solutions to the 2D Euler equations: These are derivatives of a sub-fractional order. Details can be found in our new preprint.

I was invited to give a public talk during the closing ceremony of the 10 minutes museum Mathematics in the climate crisis in the LWL natural history museum. The title of my talk was "About the climate, the hunt for a million and James Bond's martini cult"

In a new preprint with Dominik Winkler, we study the large time dynamics for the axisymmetric Navier-Stokes equations. What is particularly nice in that paper: We establish higher order asymptotics based on simple energy methods!

After three years in Münster, Víctor Navarro Fernández moved on to a new position at the Imperial College in London. Víctor had worked on stability problems for advection equations and he studied mixing in fluid flows. All the best in London!

Together with Beomjun Choi, I revisited the problem of fast diffusions on a bounded domain. More precisely, we study the solution near extinction. We construct invariant manifolds with the help of which the solutions can be approximated by finite-dimensional objects. Here is the preprint.

Our paper on the regularity of isoperimetric sets went online in the journal on the website of the Journal Calculus of Variations and Partial Differential Equations.

In a joint work with Lisa Beck and Eleonora Cinti, I have studied the optimal regularity of isoperimetric sets in the case where the perimeter functional and the volume have both a Hölder weight. The preprint is on the arXiv.

Lukas Niebel joined our working group on April 1st. In May he will defend his dissertation thesis on regularity estimates for kinetic equations at the University of Ulm. He will assist us in studying vortex rings in ideal fluids. Welcome Lukas!

My joint work with Stefano Ceci on point vortex dynamics in viscous fluids has been published in the journal Mathematische Annalen.

After having waited for the reviews for almost 2 years, my paper on bounds on the rates of enhanced dissipation was finally published in Communications in Mathematical Physics. Uff...

Together with my PhD student Dominik Winkler I investigate the large time asymptotics for solutions to a thin film equation. We achieve this by constructing invariant manifolds near the self-similar solution. Here's the preprint: https://arxiv.org/abs/2212.02262.

After having studied Random Dynamical Systems in our reading seminar Advanced Topics in ANALYSIS & PDEs for about ten months, we return to the deterministic setting and read Villani's monograph on hypocoercivity.

I am very happy that Nicola De Nitti from FAU Erlangen-Nürnberg joins us for a 2-months period. Nicola's stay is funded by the Cluster of Excellence as part of a guest program for young scientists.

I was invited to give a lecture at the public science festival Schlauraum in Münster. I chose to speak about The Mathematics of turbulent flows and the search for a million dollars.

Stefano Ceci's and my work on the motion of vortices in ideal fluids has been published in the Philosophical Transactions of the Royal Society A. Founded in 1665, this is the oldest scientific journal published in English. Our paper belongs to a theme issue entitled "Mathematical problems in physical fluid dynamics".

It is known that the transport equation with Sobolev velocity fields has poor regularity properties: Solutions may propagate only derivatives of logarithmic order. In a new work with David Meyer, we approach this problem with the help of Littlewood-Paley theory, which deals with a decomposition of the solution into dyadic frequency blocks. Our results have implications on mixing rates, rates of enhanced dissipation and on the zero-diffusivity-limit. The preprint is available here: https://arxiv.org/abs/2203.10860.

In a joint work with Stefano Ceci, I have investigated the motion of vorticity fields in two-dimensional viscous fluids. Our results show the optimal rates of spreading of vortex regions due to viscosity. This is a neat improvement on the existing literature. The motion of the vortices itself was predicted by Helmholtz in 1858! Here's the preprint: https://arxiv.org/abs/2203.07185.

I am honored to have been invited as a mini course speaker at the Summer School Horizons in non-linear PDEs in Ulm.

In a joint work with Beomjun Choi and Robert McCann I have investigated the fast diffusion equation on a bounded domain. In this model, the diffusivity becomes singular at the domain boundary, over which the mass escapes completely in finite time.  We show a nice dichotomy for the extinction rates in rescaled variables: The rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow. Here is a link to the preprint: https://arxiv.org/abs/2202.02769

Víctor Navarro-Fernández and André Schlichting have finished a new work on optimal error bounds for finite volume discretizations of advection-diffusion equations with rough coefficients. They prove that the numerical error cannot be larger than the order of the mesh size, which is much better than the purely advective case. Here is a link to the paper: https://arxiv.org/abs/2201.10411.

We resume our reading seminar. We start the year with a focus on Random Dynamical Systems. You can find more information on this webpage under Advanced Topics in ANALYSIS & PDEs.