Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior

Handle URI:
http://hdl.handle.net/10754/597773
Title:
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior
Authors:
Markowich, Peter; Lorz, Alexander; Francesco, Marco
Abstract:
We study the system ct + u · ∇c = ∇c -nf(c) nt + u · ∇n = ∇n m - ∇ · (n×(c) ∇c) ut + u·∇u + ∇P - η∇u + n∇φ/ = 0 ∇·u = 0. arising in the modelling of the motion of swimming bacteria under the effect of diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers in the literature lies in the presence of nonlinear porous-medium-like diffusion in the equation for the density n of the bacteria, motivated by a finite size effect. We prove that, under the constraint m ε (3/2, 2] for the adiabatic exponent, such system features global in time solutions in two space dimensions for large data. Moreover, in the case m = 2 we prove that solutions converge to constant states in the large-time limit. The proofs rely on standard energy methods and on a basic entropy estimate which cannot be achieved in the case m = 1. The case m = 2 is very special as we can provide a Lyapounov functional. We generalize our results to the three-dimensional case and obtain a smaller range of exponents m ε (m*, 2] with m* > 3/2, due to the use of classical Sobolev inequalities.
Citation:
Markowich P, Lorz A, Francesco M (2010) Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. DCDS-A 28: 1437–1453. Available: http://dx.doi.org/10.3934/dcds.2010.28.1437.
Publisher:
American Institute of Mathematical Sciences (AIMS)
Journal:
Discrete and Continuous Dynamical Systems
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Jun-2010
DOI:
10.3934/dcds.2010.28.1437
Type:
Article
ISSN:
1078-0947
Sponsors:
This publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). P. Markowich acknowledges support from his Royal Society Wolfson Research Merit Award. M. Di Francesco is partially supported by the Italian MIUR under the PRIN program 'Nonlinear Systems of Conservation Laws and Fluid Dynamics'. A. Lorz acknowledges support from KAUST. The authors acknowledge fruitful discussions with Christian Schmeiser and with Jose A. Carrillo. Moreover, the authors would like to thank the referees for the extremely useful comments which helped to improve the article.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorMarkowich, Peteren
dc.contributor.authorLorz, Alexanderen
dc.contributor.authorFrancesco, Marcoen
dc.date.accessioned2016-02-25T12:56:28Zen
dc.date.available2016-02-25T12:56:28Zen
dc.date.issued2010-06en
dc.identifier.citationMarkowich P, Lorz A, Francesco M (2010) Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. DCDS-A 28: 1437–1453. Available: http://dx.doi.org/10.3934/dcds.2010.28.1437.en
dc.identifier.issn1078-0947en
dc.identifier.doi10.3934/dcds.2010.28.1437en
dc.identifier.urihttp://hdl.handle.net/10754/597773en
dc.description.abstractWe study the system ct + u · ∇c = ∇c -nf(c) nt + u · ∇n = ∇n m - ∇ · (n×(c) ∇c) ut + u·∇u + ∇P - η∇u + n∇φ/ = 0 ∇·u = 0. arising in the modelling of the motion of swimming bacteria under the effect of diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers in the literature lies in the presence of nonlinear porous-medium-like diffusion in the equation for the density n of the bacteria, motivated by a finite size effect. We prove that, under the constraint m ε (3/2, 2] for the adiabatic exponent, such system features global in time solutions in two space dimensions for large data. Moreover, in the case m = 2 we prove that solutions converge to constant states in the large-time limit. The proofs rely on standard energy methods and on a basic entropy estimate which cannot be achieved in the case m = 1. The case m = 2 is very special as we can provide a Lyapounov functional. We generalize our results to the three-dimensional case and obtain a smaller range of exponents m ε (m*, 2] with m* > 3/2, due to the use of classical Sobolev inequalities.en
dc.description.sponsorshipThis publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). P. Markowich acknowledges support from his Royal Society Wolfson Research Merit Award. M. Di Francesco is partially supported by the Italian MIUR under the PRIN program 'Nonlinear Systems of Conservation Laws and Fluid Dynamics'. A. Lorz acknowledges support from KAUST. The authors acknowledge fruitful discussions with Christian Schmeiser and with Jose A. Carrillo. Moreover, the authors would like to thank the referees for the extremely useful comments which helped to improve the article.en
dc.publisherAmerican Institute of Mathematical Sciences (AIMS)en
dc.subjectChemotaxis modelen
dc.subjectNonlinear diffusionen
dc.subjectStokes equationsen
dc.titleChemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavioren
dc.typeArticleen
dc.identifier.journalDiscrete and Continuous Dynamical Systemsen
dc.contributor.institutionDepartment of Pure and Applied Mathematics, 67100 L'Aquila, Italyen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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