Mathematical Analysis of a PDE System for Biological Network Formation

DOI10.1080/03605302.2014.968792zbMath1345.35120arXiv1405.0857OpenAlexW1998862758WikidataQ60502727 ScholiaQ60502727MaRDI QIDQ5256312

Jan Haskovec, Perthame, Benoît, Peter Alexander Markowich

Publication date: 22 June 2015

Published in: Communications in Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1405.0857

zbMATH Keywords

stabilityweak solutionsnonlinear parabolic equationsenergy dissipationnetwork formationbifurcation analysis

Mathematics Subject Classification ID

Nonlinear parabolic equations (35K55) PDEs in connection with biology, chemistry and other natural sciences (35Q92) Bifurcations in context of PDEs (35B32) Systems biology, networks (92C42)

Related Items (26)

Notes on a PDE system for biological network formation ⋮ Branching structures emerging from a continuous optimal transport model ⋮ Optimisation of Biological Transport Networks ⋮ An Efficient Iterative Method for the Formulation of Flow Networks ⋮ Global existence and decay estimates of solutions of a parabolic-elliptic-parabolic system for ion transport networks ⋮ Implicit and semi-implicit numerical schemes for the gradient flow of the formation of biological transport networks ⋮ Elliptic equations with discontinuous coefficients in generalized Morrey spaces ⋮ Efficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation ⋮ Emergence of biological transportation networks as a self-regulated process ⋮ A modified and efficient phase field model for the biological transport network ⋮ Hölder continuity of weak solutions to an elliptic-parabolic system modeling biological transportation network ⋮ Global existence of strong solutions to a biological network formulation model in 2+1 dimensions ⋮ Discontinuous Galerkin methods for network patterning phase-field models ⋮ Long time behavior of the solution to a parabolic-elliptic system ⋮ Regularity theorems for a biological network formulation model in two space dimensions ⋮ Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model ⋮ Energy-production-rate preserving numerical approximations to network generating partial differential equations ⋮ Global existence and eventual smoothness in a 2-D parabolic-elliptic system arising from ion transport networks ⋮ Blowup criterion of classical solutions for a parabolic-elliptic system in space dimension 3 ⋮ Partial regularity of weak solutions to a PDE system with cubic nonlinearity ⋮ Supplementary variable method for structure-preserving approximations to partial differential equations with deduced equations ⋮ Global dynamics and zero-diffusion limit of a parabolic-elliptic-parabolic system for ion transport networks ⋮ A new model for the emergence of blood capillary networks ⋮ Classical solution of a PDE system stemming from auxin transport model for leaf venation ⋮ Murray’s law for discrete and continuum models of biological networks ⋮ Rigorous continuum limit for the discrete network formation problem

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