Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior
KAUST Grant NumberKUK-I1-007-43
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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.
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.
SponsorsThis 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.