TY - JOUR
T1 - Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior
AU - Markowich, Peter A.
AU - Lorz, Alexander
AU - Francesco, Marco
N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
PY - 2010/7/8
Y1 - 2010/7/8
N2 - 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.
AB - 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.
UR - http://hdl.handle.net/10754/597773
UR - http://aimsciences.org//article/doi/10.3934/dcds.2010.28.1437
UR - http://www.scopus.com/inward/record.url?scp=77958028070&partnerID=8YFLogxK
U2 - 10.3934/dcds.2010.28.1437
DO - 10.3934/dcds.2010.28.1437
M3 - Article
AN - SCOPUS:77958028070
VL - 28
SP - 1437
EP - 1453
JO - Discrete and Continuous Dynamical Systems
JF - Discrete and Continuous Dynamical Systems
SN - 1078-0947
IS - 4
ER -