Seminar 14.12.2017: Jan Haskovec

Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar

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Seminar 14.12.2017: Jan Haskovec

Postby quittner » Wed Dec 06, 2017 10:22 am

Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Thursday 14.12.2017 at 14:00 Lecture room M-223

Jan Haškovec (KAUST):
Discrete and continuum modeling of biological network formation

Motivated by recent papers describing rules for natural network formation
in discrete settings, we propose an elliptic-parabolic system
of partial differential equations. The model describes the pressure field
due to Darcyâs type equation and the dynamics of the conductance network
under pressure force effects with a diffusion rate representing randomness
in the material structure. After a short overview of the principles
of discrete network modeling, we will show how to derive the corresponding
macroscopic (continuum) description. The highly unusual structure
of the resulting PDE system induces several interesting challenges
for its mathematical analysis. We give a short overview of the tools
and tricks that can be used to overcome them. In particular,
we present results regarding the existence of weak solutions of the system,
based on recent results on elliptic regularity theory. Moreover, we study
the structure and stability properties of steady states that play
a central role to understand the pattern capacity of the system.
We present results of systematic numerical simulations of the system that
provide further insights into the properties of the network-type solutions.
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