Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion

Xiaoyu Chen; Jijie Zhao; Qian Zhang

doi:10.3934/dcds.2022062

Article Contents

2022, Volume 42, Issue 9: 4489-4522. Doi: 10.3934/dcds.2022062

This issue Previous Article Comparison principles for nonlocal Hamilton-Jacobi equations Next Article Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation

Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion

Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, 071002, China

^*Corresponding author: zhangqian@hbu.edu.cn
^*Corresponding author: zhangqian@hbu.edu.cn

Received: October 31, 2021

Revised: March 31, 2022

Published: August 2022

Xiaoyu Chen is supported by the Postgraduate Innovation Foundation of Hebei University [grant number HBU2022ss013], Qian Zhang is supported by the the Natural Science Foundation of Hebei Province [grant number A2020201014 and A2019201106]; the Second Batch of Young Talents of Hebei Province; Nonlinear Analysis Innovation Team of Hebei University

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we consider the Cauchy problem for the three dimensional axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion $ \Delta n^m $. Taking advantage of the structure of axisymmetric flow without swirl, we show the global existence of weak solutions for the chemotaxis-Navier-Stokes equations with $ m=\frac{5}{3} $.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 35Q92, 35Q35, 92C17.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	H. Abidi, T. Hmidi and S. Keraani, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system, Discrete Contin. Dyn. Syst., 29 (2011), 737-756. doi: 10.3934/dcds.2011.29.737.
[2]	H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.
[3]	P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0.
[4]	A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 2006 (2006), 1-33.
[5]	V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in ${\mathbb R}^2$, Commun. Math. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8.
[6]	M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled Chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297. doi: 10.3934/dcds.2013.33.2271.
[7]	A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190. doi: 10.1017/jfm.2011.534.
[8]	M. Di Francesco, A. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437.
[9]	R. J. Duan, A. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199.
[10]	R. J. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Notices, 2014 (2014), 1833-1852. doi: 10.1093/imrn/rns270.
[11]	L. C. Evans, Partial Differential Equations, 2^nd edition, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 2010. doi: 10.1090/gsm/019.
[12]	T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 27 (2010), 1227-1246. doi: 10.1016/j.anihpc.2010.06.001.
[13]	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.
[14]	J. G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.
[15]	A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507.
[16]	A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574. doi: 10.4310/CMS.2012.v10.n2.a7.
[17]	A. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002.
[18]	C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and Applications to Fluid Dynamics Equations, Monographs on Modern pure mathematics, No. 142, Science Press, Beijing, 2012.
[19]	C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. phys., 321 (2013), 33-67. doi: 10.1007/s00220-013-1721-2.
[20]	C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl., 101 (2014), 842-872. doi: 10.1016/j.matpur.2013.10.007.
[21]	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.
[22]	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.
[23]	Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.
[24]	Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.
[25]	I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near constant lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.
[26]	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.
[27]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[28]	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.
[29]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[30]	M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational 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 Dif. Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2.
[32]	M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002.
[33]	M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Am. Math. Soc., 369 (2017), 3067-3125. doi: 10.1090/tran/6733.
[34]	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.
[35]	M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339-1401. doi: 10.1016/j.jfa.2018.12.009.
[36]	Q. Zhang, Local well-posedness for the chemotaxis-Navier-Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl., 17 (2014), 89-100. doi: 10.1016/j.nonrwa.2013.10.008.
[37]	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.
[38]	Q. Zhang and P. Wang, Global well-posedness for the 2D incompressible four-component chemotaxis-Navier-Stokes equations, J. Differential Equations, 269 (2020), 1656-1692. doi: 10.1016/j.jde.2020.01.019.
[39]	Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemptaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105. doi: 10.1137/130936920.
[40]	Q. Zhang and X. Zheng, Global well-posedness of axisymmetric solution to the 3D axisymmetric chemotaxis-Navier-Stokes equations with logistic source, J. Differential Equations, 274 (2021), 576-612. doi: 10.1016/j.jde.2020.10.024.