TY - JOUR
T1 - Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion
T2 - Global existence and asymptotic behavior
AU - Di Francesco, Marco
AU - Lorz, A.
AU - Markowich, P.
PY - 2010/12/1
Y1 - 2010/12/1
N2 - We study a system 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 a system 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://www.scopus.com/inward/record.url?scp=77958028070&partnerID=8YFLogxK
UR - http://dx.doi.org/10.3934/dcds.2010.28.1437
U2 - 10.3934/dcds.2010.28.1437
DO - 10.3934/dcds.2010.28.1437
M3 - Article
AN - SCOPUS:77958028070
SN - 1078-0947
VL - 28
SP - 1437
EP - 1453
JO - Discrete and Continuous Dynamical Systems
JF - Discrete and Continuous Dynamical Systems
IS - 4
ER -