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

Abstract

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

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

It is proved that global weak solutions exist whenever and the initial data are sufficiently regular satisfying and . 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 .

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

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

On montre que les solutions faibles globales existent quand et la donnée initiale est suffisamment régulière et vérifie et . 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 .

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