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 language||English (US)|
|Number of pages||39|
|Journal||Communications in Partial Differential Equations|
|State||Published - Feb 4 2015|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: BP is (partially) funded by the french "ANR blanche" project Kibord (ANR-13-BS01-0004) and by Institut Universitaire de France. PM acknowledges support of the Fondation Sciences Mathematiques de Paris in form of his Excellence Chair 2011.