TY - JOUR

T1 - Tensor PDE model of biological network formation

AU - Haskovec, Jan

AU - Markowich, Peter A.

AU - Pilli, Giulia

N1 - KAUST Repository Item: Exported on 2022-04-14
Acknowledgements: G. P. acknowledges support from the Austrian Science Fund
(FWF) through the grants F 65 and W 1245.

PY - 2022/4/11

Y1 - 2022/4/11

N2 - 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 L2-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.

AB - 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 L2-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.

UR - http://hdl.handle.net/10754/676232

UR - https://www.intlpress.com/site/pub/pages/journals/items/cms/content/vols/0020/0004/a010/

U2 - 10.4310/cms.2022.v20.n4.a10

DO - 10.4310/cms.2022.v20.n4.a10

M3 - Article

VL - 20

SP - 1173

EP - 1191

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 4

ER -