Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion

Youshan Tao; Michael Winkler

doi:10.1016/j.anihpc.2012.07.002

JournalsaihpcVol. 30, No. 1pp. 157–178

Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion

Youshan Tao
Department of Applied Mathematics, Dong Hua University, Shanghai 200051, PR China
Michael Winkler
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

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Abstract

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-Stokes system generalizing the prototype

⎩ ⎨ ⎧ n_{t} + u \cdot \nabla n = Δ n^{m} - \nabla \cdot (n \nabla c), c_{t} + u \cdot \nabla c = Δ c - n c, u_{t} + \nabla P = Δ u + n \nabla ϕ, \nabla \cdot u = 0,

which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid.

It is proved that global weak solutions exist whenever $m > \frac{8}{7}$ and the initial data $(n_{0}, c_{0}, u_{0})$ are sufficiently regular satisfying $n_{0} > 0$ and $c_{0} > 0$ . This extends a recent result by Di Francesco, Lorz and Markowich [M. Di Francesco, A. Lorz, P.A. Markowich, Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A 28 (2010) 1437–1453] which asserts global existence of weak solutions under the constraint $m \in [\frac{7 + 217}{12}, 2]$ .

Résumé

Ce papier considère un problème aux limites dans des domaines tridimensionnels réguliers et bornés, plus précisément, un système couplé de chemotaxie-Stokes qui généralise le prototype

⎩ ⎨ ⎧ n_{t} + u \cdot \nabla n = Δ n^{m} - \nabla \cdot (n \nabla c), c_{t} + u \cdot \nabla c = Δ c - n c, u_{t} + \nabla P = Δ u + n \nabla ϕ, \nabla \cdot u = 0

et qui décrit le mouvement des bactéries nageuses conduites par lʼoxygène dans un fluide incompressible.

On montre que les solutions faibles globales existent quand $m > \frac{8}{7}$ et la donnée initiale $(n_{0}, c_{0}, u_{0})$ est suffisamment régulière et vérifie $n_{0} > 0$ et $c_{0} > 0$ . Cela étend le résultat récent de Di Francesco, Lorz et Markowich [M. Di Francesco, A. Lorz, P.A. Markowich, Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A 28 (2010) 1437–1453] qui affirme lʼexistence globale de solutions faibles sous la contrainte $m \in [\frac{7 + 217}{12}, 2]$ .

Cite this article

Youshan Tao, Michael Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 1, pp. 157–178

DOI 10.1016/J.ANIHPC.2012.07.002