Mathematical Analysis of a PDE System for Biological Network Formation

Handle URI:
http://hdl.handle.net/10754/575640
Title:
Mathematical Analysis of a PDE System for Biological Network Formation
Authors:
Haskovec, Jan; Markowich, Peter A. ( 0000-0002-3704-1821 ) ; Perthame, Benoit
Abstract:
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.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Informa UK Limited
Journal:
Communications in Partial Differential Equations
Issue Date:
4-Feb-2015
DOI:
10.1080/03605302.2014.968792
Type:
Article
ISSN:
0360-5302; 1532-4133
Sponsors:
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.
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, Benoiten
dc.date.accessioned2015-08-24T08:34:49Zen
dc.date.available2015-08-24T08:34:49Zen
dc.date.issued2015-02-04en
dc.identifier.issn0360-5302en
dc.identifier.issn1532-4133en
dc.identifier.doi10.1080/03605302.2014.968792en
dc.identifier.urihttp://hdl.handle.net/10754/575640en
dc.description.abstractMotivated 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.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 acknowledges support of the Fondation Sciences Mathematiques de Paris in form of his Excellence Chair 2011.en
dc.publisherInforma UK Limiteden
dc.titleMathematical Analysis of a PDE System for Biological Network Formationen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalCommunications in Partial Differential Equationsen
dc.contributor.institutionUniv Paris 06, Univ Paris 04, UMR 7598, Lab Jacques Louis Lions, Paris, Franceen
dc.contributor.institutionCNRS, UMR 7598, Lab Jacques Louis Lions, Paris, Franceen
kaust.authorHaskovec, Janen
kaust.authorMarkowich, Peter A.en
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