Citation: |
[1] |
X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913.doi: 10.1088/0951-7715/27/8/1899. |
[2] |
M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Continuous Dynam. Systems, 33 (2013), 2271-2297.doi: 10.3934/dcds.2013.33.2271. |
[3] |
M. Chae, K. Kang and J. Lee, Global Existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Part. Diff. Eqs., 39 (2014), 1205-1235.doi: 10.1080/03605302.2013.852224. |
[4] |
R. Duan, A. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Diff. Eqs., 35 (2010), 1635-1673.doi: 10.1080/03605302.2010.497199. |
[5] |
Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212.doi: 10.1016/0022-0396(86)90096-3. |
[6] |
Y. Giga and H. Sohr, Abstract $L^p$ estimate for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.doi: 10.1016/0022-1236(91)90136-S. |
[7] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin/Heidelberg, 1981. |
[8] |
S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.doi: 10.1016/j.jde.2014.01.028. |
[9] |
T. Li, A. Suen, M. Winkler and C. Xue, Gobal small-data solutions in a chemotaxis system with rotation, Math. Mod. Meth. Appl. Sci., 25(2015), 721-747. |
[10] |
J. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. I. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.doi: 10.1016/j.anihpc.2011.04.005. |
[11] |
J. L. Lions, Équations Différentielles Opérationnelles et Problémes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Springer, 1961. |
[12] |
A. Lorz, Coupled chemotaxis fluid equations, Math. Mod. Meth. Appl. Sci., 20 (2010), 987-1004.doi: 10.1142/S0218202510004507. |
[13] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., Providence, RI, 1968. |
[14] |
Y. Lou, Y. Tao and M. Winkler, Approching the ideal free distribution in two-species copetition models with fitness-dependent dispersal, SIAM J. Math. Anal., 46 (2014), 1228-1262.doi: 10.1137/130934246. |
[15] |
K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. |
[16] |
M. M. Porzio and V. Vespri, Hölder estimate for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.doi: 10.1006/jdeq.1993.1045. |
[17] |
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up,Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007. |
[18] |
H. Sohr, The Navier-Stokes Equations. an Elementary Functional Analytic Approach, Birkhăuser, Basel, 2001.doi: 10.1007/978-3-0348-8255-2. |
[19] |
Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.doi: 10.1016/j.jmaa.2011.02.041. |
[20] |
Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.doi: 10.1137/100802943. |
[21] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.doi: 10.1016/j.jde.2011.08.019. |
[22] |
Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.doi: 10.1016/j.jde.2011.07.010. |
[23] |
Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. I. H. Poincaré, Anal. Non Linéaire., 30 (2013), 157-178.doi: 10.1016/j.anihpc.2012.07.002. |
[24] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci., USA 102 (2005), 2277-2282. |
[25] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.doi: 10.1016/j.jde.2010.02.008. |
[26] |
M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Part. Diff. Eqs., 37 (2012), 319-351.doi: 10.1080/03605302.2011.591865. |
[27] |
M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.doi: 10.1007/s00205-013-0678-9. |
[28] |
M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, (2015), arXiv:1410.5929.doi: 10.1016/j.anihpc.2015.05.002. |
[29] |
C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math., 70 (2009), 133-167.doi: 10.1137/070711505. |
[30] |
Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.doi: 10.1137/130936920. |