Nonlinear Hyperbolic Systems of Generalized Navier-Stokes Type for Interactive Motion in Biology

Wolfgang Alt
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Wolfgang Alt: Universität Bonn, Abteilung Theoretische Biologie, Botanisches Institut

A chapter in Geometric Analysis and Nonlinear Partial Differential Equations, 2003, pp 431-461 from Springer

Abstract: Summary We review two modelling approaches to obtain genuinely nonlinear systems of one hyperbolic transport equation (for density) accompanied by parabolic or elliptic equations (for mean velocity and, eventually, pressure), namely generalized Navier-Stokes or (pseudostationary) Stokes equations. Background and applications are related to models of interactive biological motion, namely for contractile polymer networks in intra-cellular motility, for cell movement and tissue formation during wound healing as well as for cohorts of migrating birds. One approach is to derive, after suitable scaling, a formal continuum limit of (stochastic) Hamiltonian equations for ‘visco-elastic’ multi-particle networks with specific interaction laws. The other consists in the derivation of (highly) viscous two-phase flow equations by minimization of a corresponding energy-loss functional. In both procedures there remain convergence or existence problems to be solved analytically. Some results and a few numerical simulations are shown, particularly for the 1-dimensional case. For further results, technical details and for comparison with other methods we give corresponding references.

Keywords: Free Boundary; Continuum Limit; Delaunay Triangulation; Neighbor Distance; Voronoi Tesselation (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55627-2_22

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DOI: 10.1007/978-3-642-55627-2_22

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