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November  2015, 20(9): 3235-3254. doi: 10.3934/dcdsb.2015.20.3235

## Global classical solutions of a 3D chemotaxis-Stokes system with rotation

 1 School of Science, Xihua University, Chengdu 610039, China 2 Institut für Mathematik, Universität Paderborn, Paderborn 33098, Germany

Received  September 2014 Revised  December 2014 Published  September 2015

This paper considers the chemotaxis-Stokes system $$\begin{cases} \displaystyle n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\cdot\nabla c), &(x,t)\in \Omega\times (0,T),\\ \displaystyle c_t+u\cdot\nabla c=\Delta c-nc, &(x,t)\in\Omega\times (0,T),\qquad(\star)\\ \displaystyle u_t=\Delta u+\nabla P+n\nabla\phi , &(x,t)\in\Omega\times (0,T),\\ \nabla\cdot u=0,&(x,t)\in\Omega\times (0,T). \end{cases}$$ under no-flux boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^3$ with smooth boundary. Here $S$ is a matrix-valued sensitivity satisfying $|S(x,n,c)|<\tilde{C}(1+n)^{-\alpha}$ with some $\tilde{C}>0$ and $\alpha>0$. Although $(\star)$ does not possess the natural gradient-like functional structure available when $S$ reduces to a scalar function, we can still establish a new energy type inequality. Based on this inequality we achieve a coupled estimate for arbitrarily high Lebesgue norms of $n$ and $\nabla c$. This helps us to finally obtain the existence of a global classical solution when $\alpha$ is bigger than $\frac{1}{6}$.
Citation: Yulan Wang, Xinru Cao. Global classical solutions of a 3D chemotaxis-Stokes system with rotation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3235-3254. doi: 10.3934/dcdsb.2015.20.3235
##### References:
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Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops,, Comm. Part. Diff. Eqs., 37 (2012), 319. doi: 10.1080/03605302.2011.591865. Google Scholar [27] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system,, Arch. Ration. Mech. Anal., 211 (2014), 455. doi: 10.1007/s00205-013-0678-9. Google Scholar [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). doi: 10.1016/j.anihpc.2015.05.002. Google Scholar [29] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133. doi: 10.1137/070711505. Google Scholar [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. doi: 10.1137/130936920. Google Scholar

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##### References:
 [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. doi: 10.1088/0951-7715/27/8/1899. Google Scholar [2] M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations,, Discrete Continuous Dynam. Systems, 33 (2013), 2271. doi: 10.3934/dcds.2013.33.2271. Google Scholar [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. doi: 10.1080/03605302.2013.852224. Google Scholar [4] R. Duan, A. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations,, Comm. Part. Diff. Eqs., 35 (2010), 1635. doi: 10.1080/03605302.2010.497199. Google Scholar [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. doi: 10.1016/0022-0396(86)90096-3. Google Scholar [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. doi: 10.1016/0022-1236(91)90136-S. Google Scholar [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer, (1981). Google Scholar [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. doi: 10.1016/j.jde.2014.01.028. Google Scholar [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., (2015), 721. Google Scholar [10] J. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence,, Ann. I. H. Poincaré Anal. Non Linéaire, 28 (2011), 643. doi: 10.1016/j.anihpc.2011.04.005. Google Scholar [11] J. L. Lions, Équations Différentielles Opérationnelles et Problémes aux Limites,, Die Grundlehren der mathematischen Wissenschaften, (1961). Google Scholar [12] A. Lorz, Coupled chemotaxis fluid equations,, Math. Mod. Meth. Appl. Sci., 20 (2010), 987. doi: 10.1142/S0218202510004507. Google Scholar [13] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-Linear Equations of Parabolic Type,, Amer. Math. Soc. Transl., (1968). Google Scholar [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. doi: 10.1137/130934246. Google Scholar [15] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. Google Scholar [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. doi: 10.1006/jdeq.1993.1045. Google Scholar [17] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up,Global Existence and Steady States,, Birkhäuser Advanced Texts, (2007). Google Scholar [18] H. Sohr, The Navier-Stokes Equations. an Elementary Functional Analytic Approach,, Birkhăuser, (2001). doi: 10.1007/978-3-0348-8255-2. Google Scholar [19] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria,, J. Math. Anal. Appl., 381 (2011), 521. doi: 10.1016/j.jmaa.2011.02.041. Google Scholar [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. doi: 10.1137/100802943. Google Scholar [21] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692. doi: 10.1016/j.jde.2011.08.019. Google Scholar [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. doi: 10.1016/j.jde.2011.07.010. Google Scholar [23] Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion,, Ann. I. H. Poincaré, 30 (2013), 157. doi: 10.1016/j.anihpc.2012.07.002. Google Scholar [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., 102 (2005), 2277. Google Scholar [25] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [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. doi: 10.1080/03605302.2011.591865. Google Scholar [27] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system,, Arch. Ration. Mech. Anal., 211 (2014), 455. doi: 10.1007/s00205-013-0678-9. Google Scholar [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). doi: 10.1016/j.anihpc.2015.05.002. Google Scholar [29] C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133. doi: 10.1137/070711505. Google Scholar [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. doi: 10.1137/130936920. Google Scholar
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