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Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system. (English) Zbl 1378.35160

Summary: This paper concerns the coupled chemotaxis-Navier-Stokes system in the two-dimensional setting. Such a system was proposed in [I. Tuval et al., Proc. Natl. Acad. Sci. USA 102, No. 7, 2277–2282 (2005; Zbl 1277.35332)] to describe the collective effects arising in bacterial suspensions in fluid drops. Under some basic assumptions on the parameter functions \(\chi(\cdot)\), \(k(\cdot)\) and the potential function \(\phi\), which are consistent with those used by the experimentalists but weaker than those appeared in the known mathematical works, we establish the global existence of weak solutions and classical solutions for both the Cauchy problem and the initial-boundary value problem supplemented with some initial data. For the initial-boundary value problem, we also assert that the solution converges in large time to the spatially homogeneous equilibrium \((\overline{n_0}, 0, 0)\) with \(\overline{n_0} : = \frac{1}{|\Omega|} \int_{\Omega} n_0(x) \mathrm{d} x\). Our results also show that the large diffusion of the cell density or the chemical concentration can rule out the finite-time blow-up even though the Navier-Stokes fluid is included.

MSC:

35K55 Nonlinear parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35Q30 Navier-Stokes equations
92C17 Cell movement (chemotaxis, etc.)

Citations:

Zbl 1277.35332
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References:

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