

Diffusiondriven 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 nonuniform multicomponent 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 setups based on various generalizations of the CahnHilliard formulation [1, 2]. The resulting mathematical models
exhibit rather complex dynamic behaviour, multiscale in space and time, fluctuationdriven, and, the most spectacular effect, regardless initial composition
their largetime distributions in space get nonuniform (typically separated in different fixedvalue 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 networklike 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 largetime 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: 291376.
[2] N. Kenmochi and M. Niezgódka, (1994) J. Math. Anal. Appl., 188: 651679.
[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




