Oberseminar Dynamische Systeme:
Mathematische Grundlagen und Anwendungen

Sommersemester 2016
Montag, 18.4.16, 16 Uhr c.t., Raum MI 03.06.011:
B.Sc. Sebastian Günwald (TUM)
Über die Begründung der Arithmetik bei Gottlob Frege

Montag, 25.4.16, 16 Uhr c.t., Raum MI 03.06.011:
Dr. Marie-Therese Wolfram (University of Warwick, UK)
Analysis of a cross-diffusion model with excluded volume effects

Montag, 23.5.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Thomas Hagen (University of Memphis, USA)
Fluid Dynamics of Beetle Juice

Abstract: In this lecture I will introduce the interested participant to the dynamics of surface-tension driven flows of liquids in networks of channels. The origins of interest in these flows lie with the study of palm beetles and their clever defense mechanism which consists of excreting oily liquids and maintaining strong ground suction by controlling the resultant liquid bridges. Needless to say, a couple of engineers earned a lot of money by exploiting this mechanism to build a novel suction device.

These channel flows are driven by surface-tension induced volume scavenging where fluid droplets of varying sizes leech off one another to increase in volume. They are modeled by a system of nonlinear ODEs when one exploits the relationship of pressure gradient and flow rate in form of the Hagen-Poiseuille law or similar such law. Of interest in this presentation are the stability and long-term behavior of such flows in the case of power-law liquids where equilibria are non-hyperbolic.

The analytical techniques used to study some basic aspects of these fluid flows are neat, even though they fall largely within the context of "elementary math" and hardly exceed what bachelor's students know. Hence this presentation is intended as a mathematical excursion for any interested participant who wants to spend some quality time on applied mathematics and see a physics-based example where nice results can be had without brute force analysis. I'll even show some moving pictures and address some related questions for which the answers are "physically intuitive," but mathematically unproven.

The presentation is based on joint work with Paul Steen (Cornell).

Mittwoch, 25.5.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Manuel Victor Gnann (University of Michigan, USA)
Well-posedness and regularity for a thin-film free boundary problem

Abstract: Our interest lies in understanding a free boundary problem to a fourth-order thin-film equation with quadratic mobility and a zero contact angle at the triple junction, where air, liquid, and solid meet. This equation can be derived from the Navier-Stokes system with Navier-slip at the liquid-solid interface, removing the contact-line singularity that occurs if no slip is assumed. While for linear mobility (Dary dynamics) a strong analogy to the second-order porous medium equation is valid, this is not the case anymore in our setting, leading to singular expansions of solutions at the free boundary.

I will first discuss the model problem of source-type self-similar solutions, where ODE and dynamical systems theory are available, to characterize the contact-line singularity. This is based on two publications with Lorenzo Giacomelli and Felix Otto, and with Fethi Ben Belgacem and Christian Kühn, respectively. Then I will present a well-posedness result for solutions close to a traveling wave (joint with Lorenzo Giacomelli, Hans Knüpfer, and Felix Otto). I will further discuss how to obtain regularity of solutions at the free boundary, which one may view as a generalized smoothing property. The talk is concluded by outlooks on how to potentially treat general mobilities and the multi-dimensional thin-film problem.

Montag, 30.5.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Johannes Zimmer (University of Bath, UK)
Coherent motion for interacting particles: waves in the Frenkel-Kontorova chain

Abstract: In 1939, Frenkel and Kontorova proposed a model for the motion of a dislocation (an imperfection in a crystal). The model is simple, a chain of atoms following Newton's equation of motion. The atoms interact with their nearest neighbours via a harmonic spring and are exposed to a periodic (non-convex) on-site potential. Despite the simplicity, the model has proved to be a mathematical challenge. Iooss and Kirchgässner made a fundamental contribution regarding the existence of small solutions, using centre manifold theory. The talk will introduce the model and then present recent results for the coherent motion of dislocations with periodic, and possibly anharmonic, on-site potentials. For anharmonic wave-trains, the proof establishes the existence of possibly large wave-trains via centre manifold theory and then employs a fixed point argument to show the existence of a travelling dislocation. These results are joint work with Boris Buffoni (Lausanne) and Hartmut Schwetlick (Bath).

Montag, 13.6.16, 16 Uhr c.t., Raum MI 03.06.011:
M.Sc. Ines Assum (TUM)
Bifurcation Analysis of Quorum Sensing Systems with Heterogeneous Parameters

Montag, 4.7.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Bruce van-Brunt (Massey University, New Zealand)
On a functional differential equation arising in a cell division model

Montag, 18.7.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Florian Rupp (German University of Technology in Oman)
Mean-Square-Lyapunov-Basins for Random Ordinary Differential Equations

Abstract: We discuss the construction of local Lyapunov functions for asymptotically stable equilibria in dynamical systems generated by random and stochastic differential equations. A special focus is given to the conversion of systems of stochastic differential equations to equivalent systems of random differential equations, and the special stability notion in the sense of mean-square concepts that is tailored for computational considerations and Monte-Carlo approaches.