# American Institute of Mathematical Sciences

February 2019, 24(2): 831-849. doi: 10.3934/dcdsb.2018209

## Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 3 College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  October 2017 Revised  March 2018 Published  June 2018

Fund Project: The first author is partially supported by graduate research and innovation foundation of Chongqing, China (Grant No. CYB 17040). The second author is partially supported by NSFC (Grant Nos. 11771062, 11571062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826). The third author is partially supported by National Science Foundation of China (Grant Nos. 11601053, 11526042). The fourth author is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK1801059)

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source
 $\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$
in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills
 $\begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$
with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.
Citation: Dan Li, Chunlai Mu, Pan Zheng, Ke Lin. Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 831-849. doi: 10.3934/dcdsb.2018209
##### References:
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##### References:
 [1] K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logstic source, C. R. Acad. Sci. Paris. Ser. I, 351 (2013), 585-591. doi: 10.1016/j.crma.2013.07.027. [2] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. [3] 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. [4] T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080. [5] T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851. doi: 10.1016/j.jde.2012.01.045. [6] T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear system Keller-Segel system and applications to volume filling models, J. Differnetial Equations, 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004. [7] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemoatxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009. [8] K. Djie and M. Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045. [9] K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684. doi: 10.1016/j.jmaa.2014.11.045. [10] M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683. [11] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [12] T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 281-301. doi: 10.1006/aama.2001.0721. [13] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363. [14] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [15] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Dtsch. Math.- Ver., 105 (2003), 103-165. [16] S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes system with position-dependent sensitivity in 2d bounded domains, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 3463-3482. doi: 10.3934/dcds.2015.35.3463. [17] 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. [18] S. Ishida and T. Yokota, Blow-up finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569. [19] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [20] A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), 115609, 9pp. doi: 10.1063/1.4742858. [21] A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879. [22] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. [23] H. A. Levine and B. D. Sleeman, A system of reaction-diffusion equations arising in the theory of reinforced randomw walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106. [24] T. Li, A. Suen, M. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746. doi: 10.1142/S0218202515500177. [25] X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D KellerSegel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910. doi: 10.4310/CMS.2016.v14.n7.a5. [26] X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal., 37 (2017), 14-30. doi: 10.1016/j.nonrwa.2017.02.005. [27] J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel- (Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028. [28] J. Liu and Y. Wang, Boundedness and decay property in a three-dimensionlal Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999. doi: 10.1016/j.jde.2016.03.030. [29] T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. [30] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X. [31] K. J. Painter and T. 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