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doi: 10.3934/dcds.2022062
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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

Received  November 2021 Revised  April 2022 Early access April 2022

Fund Project: 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

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} $.

Citation: Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022062
References:
[1]

H. AbidiT. 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. BlanchetJ. 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. ChaeK. 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. ChertockK. FellnerA. KurganovA. 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 FrancescoA. 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. DuanA. 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, 2nd 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. MiaoJ. 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. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. 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.

show all references

References:
[1]

H. AbidiT. 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. BlanchetJ. 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. ChaeK. 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. ChertockK. FellnerA. KurganovA. 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 FrancescoA. 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. DuanA. 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, 2nd 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. MiaoJ. 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. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. 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.

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