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Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

Diffusion-driven phase separation in binary systems over complex geometries: a qualitative study of dynamical developments
Marek Niezgódka (ICM University of Warsaw) and Willi Jäger (IWR, University of Heidelberg)
Phase separation refers to a class of developments in non-uniform multi-component systems that result in spatial structure coarsening and formation of subdomains with (nearly) uniformly distributed composition. Driven by diffusive mechanisms, those phenomena require a substantial generalization of standard models, ending up with set-ups based on various generalizations of the Cahn-Hilliard formulation [1, 2]. The resulting mathematical models exhibit rather complex dynamic behaviour, multiscale in space and time, fluctuation-driven, and, the most spectacular effect, regardless initial composition their large-time distributions in space get non-uniform (typically separated in different fixed-value subdomains). Such a type of developments is characteristic of diverse classes of systems, in particular including polymers, with various extensions of the classical Flory model [4]. To the time, almost exclusively any rigorous mathematical analysis of the resulting models addressed systems evolving over geometric domains of high regularity. On the contrary, for the problems of real applied value those domains often show high irregularity, either due to low smoothness of their boundaries or caused by a variation of those domains in time up to deformations and topological changes. In formulations arising in the context of biological applications, of special interest are the domains of network-like shapes, with bulk subdomains connected by possibly quite thin channels.

The specific applications to be studied as reference problems refer to:

(1) the cardiovascular system, its biomechanics and biochemistry in the context of thrombosis developments and their clinical treatment [3],
(2) individually controlled drug delivery to selected organs, accounting distributed nature of the process dynamics.

The Ph.D. project will primarily focus on the qualitative analysis of system evolution for large times, structure of the reachable equilibria and other properties characteristic for the dynamical system. Due to high nonlinearity and multiscale nature of the systems arising at modelling, the resulting dynamical systems exhibit complex behaviour that requires novel mathematical concepts to be developed for their approximate treatment. In particular, the impact of specific domain geometries on large-time system behaviour will be explored. The study will be conducted subject to model parameters and the geometry assumed to be only approximately known.

[1] H. W. Alt and I. Pawlow, (1996) Adv. Math. Sci. Appl., 6: 291-376.
[2] N. Kenmochi and M. Niezgódka, (1994) J. Math. Anal. Appl., 188: 651-679.
[3] J. Mizerski, Ph.D. (2004) under supervision of Prof. Dr. hab. Zbigniew Religa.
[4] M. Gokieli, Ph.D. (2002) under supervision of Prof. F. Simondon and Prof. M. Niezgódka