Notes on a PDE system for biological network formation

Handle URI:
http://hdl.handle.net/10754/596892
Title:
Notes on a PDE system for biological network formation
Authors:
Haskovec, Jan; Markowich, Peter A. ( 0000-0002-3704-1821 ) ; Perthame, Benoît; Schlottbom, Matthias ( 0000-0002-2527-6498 )
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.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Notes on a PDE system for biological network formation 2016 Nonlinear Analysis: Theory, Methods & Applications
Publisher:
Elsevier BV
Journal:
Nonlinear Analysis: Theory, Methods & Applications
Issue Date:
22-Jan-2016
DOI:
10.1016/j.na.2015.12.018
Type:
Article
ISSN:
0362546X
Sponsors:
BP is (partially) funded by the french “ANR blanche” project Kibord: “ANR-13-BS01-0004” and by Institut Universitaire de France. PM and JH are supported by KAUST baseline funds and grant no. 1000000193. MS acknowledges support by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse.
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0362546X15004344
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorHaskovec, Janen
dc.contributor.authorMarkowich, Peter A.en
dc.contributor.authorPerthame, Benoîten
dc.contributor.authorSchlottbom, Matthiasen
dc.date.accessioned2016-02-22T07:21:05Zen
dc.date.available2016-02-22T07:21:05Zen
dc.date.issued2016-01-22en
dc.identifier.citationNotes on a PDE system for biological network formation 2016 Nonlinear Analysis: Theory, Methods & Applicationsen
dc.identifier.issn0362546Xen
dc.identifier.doi10.1016/j.na.2015.12.018en
dc.identifier.urihttp://hdl.handle.net/10754/596892en
dc.description.abstractWe 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.en
dc.description.sponsorshipBP is (partially) funded by the french “ANR blanche” project Kibord: “ANR-13-BS01-0004” and by Institut Universitaire de France. PM and JH are supported by KAUST baseline funds and grant no. 1000000193. MS acknowledges support by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0362546X15004344en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Nonlinear Analysis: Theory, Methods & Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Nonlinear Analysis: Theory, Methods & Applications, 22 January 2016. DOI: 10.1016/j.na.2015.12.018en
dc.subjectNetwork formationen
dc.subjectPenalty methoden
dc.subjectNumerical experimentsen
dc.titleNotes on a PDE system for biological network formationen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalNonlinear Analysis: Theory, Methods & Applicationsen
dc.eprint.versionPost-printen
dc.contributor.institutionSorbonne Universités, UPMC Univ Paris 06, Inria, Laboratoire Jacques-Louis Lions UMR CNRS 7598, F-75005, Paris, Franceen
dc.contributor.institutionInstitute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149 Münster, Germanyen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorHaskovec, Janen
kaust.authorMarkowich, Peter A.en
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.