Notes on a PDE system for biological network formation

Jan Haskovec, Peter A. Markowich, Benoît Perthame, Matthias Schlottbom

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

We present new analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field using a Darcy’s type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of Haskovec et al. (2015) regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or break-down of solutions in the spatially one-dimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameter values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.
Original languageEnglish (US)
Pages (from-to)127-155
Number of pages29
JournalNonlinear Analysis: Theory, Methods & Applications
Volume138
DOIs
StatePublished - Jan 23 2016

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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