Abstract
We study an elliptic-parabolic system of partial differential equations
describing formation of biological network structures. The model takes into
consideration the evolution of the permeability tensor under the influence of a
diffusion term, representing randomness in the material structure, a decay term
describing metabolic cost and a pressure force. A Darcy's law type equation
describes the pressure field. In the spatially two-dimensional setting, we
present a constructive, formal derivation of the PDE system from the discrete
network formation model in the refinement limit of a sequence of unstructured
triangulations. Moreover, we show that the PDE system is a formal
$L^2$-gradient flow of an energy functional with biological interpretation, and
study its convexity properties. For the case when the energy functional is
convex, we construct unique global weak solutions of the PDE system. Finally,
we construct steady state solutions in one- and multi-dimensional settings and
discuss their stability properties.