A coupled chemotaxis-fluid model: Global existence
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 5, p. 643-652
We consider a model arising from biology, consisting of chemotaxis equations coupled to viscous incompressible fluid equations through transport and external forcing. Global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the chemotaxis–Navier–Stokes system in two space dimensions, we obtain global existence for large data. In three space dimensions, we prove global existence of weak solutions for the chemotaxis–Stokes system with nonlinear diffusion for the cell density.
Nous considérons un modèle provenant de la biologie, composé dʼéquations de chimiotactisme couplées aux équations de fluide visqueux incompressible par le transport et le forçage externe. Lʼexistence globale des solutions du problème de Cauchy est étudiée sous certaines conditions. Précisément, pour le système chimiotactisme–Navier–Stokes en deux dimensions dʼespace, nous obtenons lʼexistence globale pour des données grandes. En trois dimensions dʼespace, nous démontrons lʼexistence globale des solutions faibles pour le système chimiotactisme–Stokes avec une diffusion non-linéaire de la densité des cellules.
@article{AIHPC_2011__28_5_643_0,
     author = {Liu, Jian-Guo and Lorz, Alexander},
     title = {A coupled chemotaxis-fluid model: Global existence},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {5},
     year = {2011},
     pages = {643-652},
     doi = {10.1016/j.anihpc.2011.04.005},
     zbl = {1236.92013},
     mrnumber = {2838394},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_5_643_0}
}
Liu, Jian-Guo; Lorz, Alexander. A coupled chemotaxis-fluid model: Global existence. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 5, pp. 643-652. doi : 10.1016/j.anihpc.2011.04.005. http://www.numdam.org/item/AIHPC_2011__28_5_643_0/

[1] A. Blanchet, J.A. Carrillo, N. Masmoudi, Infinite time aggregation for the critical Patlak–Keller–Segel model in 2 , Comm. Pure Appl. Math. 61 (2008), 1449-1481 | MR 2436186 | Zbl 1155.35100

[2] A. Blanchet, J. Dolbeault, B. Perthame, Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006 (2006), 1-33 | MR 2226917 | Zbl 1112.35023

[3] R. Caflisch, G. Papanicolaou, Dynamic theory of suspensions with Brownian effects, SIAM J. Appl. Math. 43 (1983), 885-906 | MR 709743 | Zbl 0543.76133

[4] V. Calvez, J. Carrillo, Volume effects in the Keller–Segel model: energy estimates preventing blow-up, J. Math. Pures Appl. 86 (2006), 155-175 | MR 2247456 | Zbl 1116.35057

[5] V. Calvez, L. Corrias, The parabolic–parabolic Keller–Segel model in 2 , Commun. Math. Sci. 6 (2008), 417-447 | MR 2433703 | Zbl 1149.35360

[6] J. Carrillo, T. Goudon, Stability and asymptotic analysis of a fluid–particle interaction model, Comm. Partial Differential Equations 31 (2006), 1349-1379 | MR 2254618 | Zbl 1105.35088

[7] F. Chalub, Y. Dolak-Struss, P. Markowich, D. Oelz, C. Schmeiser, A. Soreff, Model hierarchies for cell aggregation by chemotaxis, Math. Models Methods Appl. Sci. 16 (2006), 1173-1197 | MR 2250124 | Zbl 1094.92009

[8] R.-J. Duan, A. Lorz, P. Markowich, Global solutions to the coupled chemotaxis–fluid equations, Comm. Partial Differential Equations 35 no. 9 (2010), 1635-1673 | MR 2754058 | Zbl 1275.35005

[9] M.D. Francesco, A. Lorz, P. Markowich, Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A 28 no. 4 (2010), 1437-1453 | MR 2679718 | Zbl 1276.35103

[10] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov–Stokes equations, Japan J. Indust. Appl. Math. 15 (1998), 51-74 | MR 1610309 | Zbl 1306.76052

[11] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl. 305 (2005), 566-588 | MR 2130723 | Zbl 1065.35063

[12] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Volume 1: Incompressible Models, Oxford Lecture Ser. Math. Appl. vol. 3 (1996)

[13] A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci. 20 no. 6 (2010), 987-1004 | MR 2659745 | Zbl 1191.92004

[14] I. Tuval, L. Cisneros, C. Dombrowski, C.W. Wolgemuth, J.O. Kessler, R.E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005), 2277-2282 | Zbl 1277.35332