American Institute of Mathematical Sciences

Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux

 1 School of Mathematics, Southeast University, Nanjing 210096, China 2 Institute for Applied Mathematics, School of Mathematics, Southeast University, Nanjing 211189, China

* Corresponding author

Received  June 2018 Revised  November 2018 Published  April 2019

Fund Project: The authors are supported in part by NSF of China (No. 11671079, No. 11701290, No. 11601127, No. 11871147 and No. 11871148), and NSF of Jiangsu Province (No. BK20170896)

In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux given by
 $\begin{eqnarray*} \left\{\begin{array}{lll}n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(uS(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0,\\[1mm]c_t+u\cdot\nabla c=\Delta c-c+n,&x\in\Omega,\ \ t>0,\\[1mm]u_t+k(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,&x\in\Omega,\ \ t>0\\[1mm]\nabla\cdot u=0,&x\in\Omega,\ \ t>0 \end{array}\right.\end{eqnarray*}$
in a bounded domain
 $\Omega\subset\mathbb{R}^3$
, where
 $k\in\mathbb{R}$
,
 $\phi\in W^{2,\infty}(\Omega)$
and the given tensor-valued function
 $S$
:
 $\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}$
satisfies
 $|S(x,n,c)|\leq S_0(n+1)^{-\alpha}\ \ {\rm for\ all}\ x\in\mathbb{R}^3,\ n\geq0,\ c\geq0.$
Imposing no restriction on the size of the initial data, we establish the global existence of a very weak solution while assuming
 $m+\alpha>\frac{4}{3}$
and
 $m>\frac{1}{3}$
.
Citation: Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019064
References:
 [1] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. [2] 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. [3] R. Duan, X. Li and Z. Xiang, Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system, J. Differential Equations, 263 (2017), 6284-6316. doi: 10.1016/j.jde.2017.07.015. [4] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. [5] H. He and Q. Zhang, Global existence of weak solutions for the 3D chemotaxis-Navier-Stokes equations, Nonlinear Anal. Real World Appl., 35 (2017), 336-349. doi: 10.1016/j.nonrwa.2016.11.006. [6] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301. doi: 10.1006/aama.2001.0721. [7] 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. [8] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [9] 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. [10] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588. doi: 10.1016/j.jmaa.2004.12.009. [11] J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109. doi: 10.1142/S021820251640008X. [12] Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84. doi: 10.1016/j.na.2014.05.021. [13] Y. Li and Y. Li, Global boundedness of solutions for the chemotaxis-Navier-Stokes system in $\Bbb{R}^2$, J. Differential Equations, 261 (2016), 6570-6613. doi: 10.1016/j.jde.2016.08.045. [14] 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. [15] J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305. doi: 10.1016/j.jde.2017.01.024. [16] J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005. [17] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [18] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [19] Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Z. Angew. Math. Phys., 68 (2017), Art. 68, 26pp. doi: 10.1007/s00033-017-0816-6. [20] Z. Tan and X. Zhang, Decay estimates of the coupled chemotaxis-fluid equations in R3, J. Math. Anal. Appl., 410 (2014), 27-38. doi: 10.1016/j.jmaa.2013.08.008. [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, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183. doi: 10.3934/dcdsb.2015.20.3165. [23] 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 Natl Acad Sci U S A, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. [24] Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780. doi: 10.1142/S0218202517500579. [25] Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254. doi: 10.3934/dcdsb.2015.20.3235. [26] Y. Wang, M. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 421-466. [27] 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. [28] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. [29] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865. [30] 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. [31] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2. [32] M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115. doi: 10.1137/140979708. [33] M. Winkler, Global weak solutions in a three-dimensional chemotaxis–Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002. [34] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125. doi: 10.1090/tran/6733. [35] M. Winkler, Does fluid interaction affect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?, J. Math. Fluid. Mech., 20 (2018), 1889-1909. doi: 10.1007/s00021-018-0395-0. [36] M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151. doi: 10.1016/j.jde.2018.01.027. [37] D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444. doi: 10.1017/S0308210500004649. [38] M. Yang, Global solutions to Keller-Segel-Navier-Stokes equations with a class of large initial data in critical Besov spaces, Math. Methods Appl. Sci., 40 (2017), 7425-7437. doi: 10.1002/mma.4538. [39] H. Yu, W. Wang and S. Zheng, Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770. doi: 10.1016/j.jmaa.2017.12.048. [40] Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63. doi: 10.1016/j.jmaa.2014.03.084. [41] Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484. doi: 10.1007/s00033-015-0532-z. [42] Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2527-2551. doi: 10.3934/dcdsb.2015.20.2751. [43] Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4. [44] Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754. doi: 10.1016/j.jde.2015.05.012. [45] 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. [46] J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629. doi: 10.1016/j.jde.2017.04.005.

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References:
 [1] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. [2] 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. [3] R. Duan, X. Li and Z. Xiang, Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system, J. Differential Equations, 263 (2017), 6284-6316. doi: 10.1016/j.jde.2017.07.015. [4] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. [5] H. He and Q. Zhang, Global existence of weak solutions for the 3D chemotaxis-Navier-Stokes equations, Nonlinear Anal. Real World Appl., 35 (2017), 336-349. doi: 10.1016/j.nonrwa.2016.11.006. [6] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301. doi: 10.1006/aama.2001.0721. [7] 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. [8] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [9] 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. [10] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588. doi: 10.1016/j.jmaa.2004.12.009. [11] J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109. doi: 10.1142/S021820251640008X. [12] Y. Li and Y. Li, Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84. doi: 10.1016/j.na.2014.05.021. [13] Y. Li and Y. Li, Global boundedness of solutions for the chemotaxis-Navier-Stokes system in $\Bbb{R}^2$, J. Differential Equations, 261 (2016), 6570-6613. doi: 10.1016/j.jde.2016.08.045. [14] 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. [15] J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305. doi: 10.1016/j.jde.2017.01.024. [16] J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005. [17] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [18] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [19] Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Z. Angew. Math. Phys., 68 (2017), Art. 68, 26pp. doi: 10.1007/s00033-017-0816-6. [20] Z. Tan and X. Zhang, Decay estimates of the coupled chemotaxis-fluid equations in R3, J. Math. Anal. Appl., 410 (2014), 27-38. doi: 10.1016/j.jmaa.2013.08.008. [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, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183. doi: 10.3934/dcdsb.2015.20.3165. [23] 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 Natl Acad Sci U S A, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. [24] Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780. doi: 10.1142/S0218202517500579. [25] Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254. doi: 10.3934/dcdsb.2015.20.3235. [26] Y. Wang, M. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 421-466. [27] 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. [28] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. [29] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865. [30] 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. [31] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2. [32] M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115. doi: 10.1137/140979708. [33] M. Winkler, Global weak solutions in a three-dimensional chemotaxis–Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002. [34] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125. doi: 10.1090/tran/6733. [35] M. Winkler, Does fluid interaction affect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?, J. Math. Fluid. Mech., 20 (2018), 1889-1909. doi: 10.1007/s00021-018-0395-0. [36] M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151. doi: 10.1016/j.jde.2018.01.027. [37] D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444. doi: 10.1017/S0308210500004649. [38] M. Yang, Global solutions to Keller-Segel-Navier-Stokes equations with a class of large initial data in critical Besov spaces, Math. Methods Appl. Sci., 40 (2017), 7425-7437. doi: 10.1002/mma.4538. [39] H. Yu, W. Wang and S. Zheng, Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770. doi: 10.1016/j.jmaa.2017.12.048. [40] Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63. doi: 10.1016/j.jmaa.2014.03.084. [41] Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484. doi: 10.1007/s00033-015-0532-z. [42] Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2527-2551. doi: 10.3934/dcdsb.2015.20.2751. [43] Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4. [44] Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754. doi: 10.1016/j.jde.2015.05.012. [45] 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. [46] J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629. doi: 10.1016/j.jde.2017.04.005.
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