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) no. 1, p. 157-178
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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·n=Δn m -·(nc),c t +u·c=Δc-nc,u t +P=Δu+nφ,·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>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[7+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 {n t +u·n=Δn m -·(nc),c t +u·c=Δc-nc,u t +P=Δu+nφ,·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>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[7+217 12,2].
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},
     publisher = {Elsevier},
     volume = {30},
     number = {1},
     year = {2013},
     pages = {157-178},
     doi = {10.1016/j.anihpc.2012.07.002},
     zbl = {1283.35154},
     mrnumber = {3011296},
     language = {en},
     url = {http://www.numdam.org/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) no. 1, pp. 157-178. doi : 10.1016/j.anihpc.2012.07.002. http://www.numdam.org/item/AIHPC_2013__30_1_157_0/

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