Athanasios Stylianou (Univ. Kassel): Pattern formation in systems at equilibrium


Abstract:  The formation of patterns has mostly been investigated in systems driven far from equilibrium (e.g. Rayleigh–Bénard convection, Taylor–Couette flow, current instabilities, morphogenesis etc), whereas, patterns in conservative stationary states (e.g. shell buckling, surface instabilities in dielectric liquids etc), although of similar importance, have not received as much systematic attention. In this talk two problems that are closely related will be presented, as well as corresponding techniques that are used in order to understand the appearance of patterns in each case.

The first problem under consideration is the so-called Rosensweig instability: placing a ferrofluid in a vertical magnetic field and surpassing a critical value, leads to the emergence of static liquid waves on the surface. The mathematical model of this phenomenon is a free-boundary problem, which has an inherently different structure than the usual case-study: the functional under consideration is not convex, but of concave-convex type. As a result, standard techniques are not directly applicable. Results on small-amplitude waves will be presented, as well as an approach that leads to a more general theory in the spirit of De Giorgi-Almgren.

The second problem involves a similar phenomenon, but in a dipolar Bose-Einstein condensate: Recent observations have been made during experiments with dysprosium, not accounted for by the standard mean field theory. These experiments produced a stable droplet crystal, similar to ones observed in classical ferrofluids. In contrast to the observation, mean field theory predicted the collapse of these droplets to extremely high densities. A rigorous existence theory for the state-of-the-art mathematical model will be presented, as well as an approach to model and prove the emergence of the aforementioned stable pattern.


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