Mathematical Analysis of a PDE System for Biological Network Formation

Jan Haskovec, Peter A. Markowich, Benoit Perthame

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


Motivated by recent physics papers describing rules for natural network formation, we study an elliptic-parabolic system of partial differential equations proposed by Hu and Cai [13, 15]. The model describes the pressure field thanks to Darcy's type equation and the dynamics of the conductance network under pressure force effects with a diffusion rate D >= 0 representing randomness in the material structure. We prove the existence of global weak solutions and of local mild solutions and study their long term behavior. It turns out that, by energy dissipation, steady states play a central role to understand the network formation capacity of the system. We show that for a large diffusion coefficient D, the zero steady state is stable, while network formation occurs for small values of D due to the instability of the zero steady state, and the borderline case D = 0 exhibits a large class of dynamically stable (in the linearized sense) steady states.
Original languageEnglish (US)
Pages (from-to)918-956
Number of pages39
JournalCommunications in Partial Differential Equations
Issue number5
StatePublished - Feb 4 2015


Dive into the research topics of 'Mathematical Analysis of a PDE System for Biological Network Formation'. Together they form a unique fingerprint.

Cite this