Decay estimates of the coupled chemotaxis-fluid equations in \(R^3\). (English) Zbl 1333.92015
Summary: In this paper, we are concerned with a chemotaxis-Navier-Stokes model, arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations with transport and external force. The optimal convergence rates of classical solutions to the chemotaxis-Navier-Stokes system for small initial perturbation around constant states are obtained by pure energy method under the assumption the initial data belong to \(\dot H^{-s}\cap H^N\), \(N\geqslant 3\) \((0\leqslant s<3/2)\). The \(\dot H^{-s}\) \((0\leqslant s<3/2)\) negative Sobolev norms are shown to be preserved along time evolution. Compared to the result in [R. Duan et al., Commun. Partial Differ. Equations 35, No. 9, 1635–1673 (2010; Zbl 1275.35005)], we obtain the optimal decay rates of the higher-order spatial derivatives of the solutions.
MSC:
| 92C17 | Cell movement (chemotaxis, etc.) |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
Navier-Stokes equations; chemotaxis; chemotaxis-fluid interaction; energy method; optimal decay rates; Sobolev interpolationCitations:
Zbl 1275.35005References:
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