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

Tao, Youshan; Winkler, Michael

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 1, pp. 157-178.

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

$$\{\begin{array}{c}{n}_t+u·\nabla n=\Delta n^m-\nabla ·\left(n\nabla c\right),\hfill \\ c_t+u·\nabla c=\Delta c-nc,\hfill \\ u_t+\nabla P=\Delta u+n\nabla \phi ,\hfill \\ \nabla ·u=0\hfill \end{array}$$

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 $m\in [\frac{7+\sqrt{217}}{12},2]$.

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

$$\{\begin{array}{c}{n}_t+u·\nabla n=\Delta n^m-\nabla ·\left(n\nabla c\right),\hfill \\ c_t+u·\nabla c=\Delta c-nc,\hfill \\ u_t+\nabla P=\Delta u+n\nabla \phi ,\hfill \\ \nabla ·u=0,\hfill \end{array}$$

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 $m\in [\frac{7+\sqrt{217}}{12},2]$.

Classification : 35K55, 35Q92, 35Q35, 92C17
Keywords: Chemotaxis, Stokes, Nonlinear diffusion, Global existence, Boundedness
Mots clés : Chemotaxie, Stokes, Diffusion nonlinéaire, Existence globale, Estimation uniforme

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