Mathematical Analysis of a PDE System for Biological Network Formation

作者：Haskovec Jan^*; Markowich Peter; Perthame Benoit

来源：Communications in Partial Differential Equations, 2015, 40(5): 918-956.

DOI：10.1080/03605302.2014.968792

• 92C42
• 35B32
• 35K55
• Bifurcation analysis
• Network formation
• Stability
• Energy dissipation
• Weak solutions

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.

• 出版日期2015-5-4