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), p. 157-178 / Harvested from Numdam

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Résumé
Abstract

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 ${\begin{matrix} n_{t} + u \cdot \nabla n = Δ n^{m} - \nabla \cdot (n \nabla c), \\ c_{t} + u \cdot \nabla c = Δ c - n c, \\ u_{t} + \nabla P = Δ u + n \nabla φ, \\ \nabla \cdot u = 0 \end{matrix}$ 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 ${\begin{matrix} n_{t} + u \cdot \nabla n = Δ n^{m} - \nabla \cdot (n \nabla c), \\ c_{t} + u \cdot \nabla c = Δ c - n c, \\ u_{t} + \nabla P = Δ u + n \nabla φ, \\ \nabla \cdot u = 0, \end{matrix}$ 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 $m \in [\frac{7 + \sqrt{217}}{12}, 2]$ .

Publié le : 2013-01-01

MR 3011296 | Zbl 1283.35154

DOI : https://doi.org/10.1016/j.anihpc.2012.07.002
Classification: 35K55, 35Q92, 35Q35, 92C17
Mots clés: Chemotaxie, Stokes, Diffusion nonlinéaire, Existence globale, Estimation uniforme

@article{AIHPC_2013__30_1_157_0,
     author = {Tao, Youshan and Winkler, Michael},
     title = {Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {157-178},
     doi = {10.1016/j.anihpc.2012.07.002},
     mrnumber = {3011296},
     zbl = {1283.35154},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_1_157_0}
}

Tao, Youshan; Winkler, Michael. Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 157-178. doi : 10.1016/j.anihpc.2012.07.002. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_1_157_0/

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