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Global solutions to the coupled chemotaxis-fluid equations. (English) Zbl 1275.35005

Summary: We are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the chemotaxis-Navier-Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.

MSC:

35A01 Existence problems for PDEs: global existence, local existence, non-existence
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35Q35 PDEs in connection with fluid mechanics
92C17 Cell movement (chemotaxis, etc.)
35B45 A priori estimates in context of PDEs
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