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June  2019, 39(6): 3413-3441. doi: 10.3934/dcds.2019141

Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion

The School of Mathematics and System Science, Beihang University, Beijing 100191, China

* Corresponding author: Xiaoxin Zheng

Received  August 2018 Revised  December 2018 Published  February 2019

In this paper, we study Cauchy problem of the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Taking advantage of a coupling structure of the equations and using the damping effect of the growth term $ g(n) $, we obtain the necessary estimates of solution $ (n,c,u) $ without the diffusion term $ \Delta n $. These uniform estimates enable us to establish the global-in-time existence of almost weak solutions for the system.

Citation: Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled Chemotaxis-fluid equations, Discrete Continous Dynam. Systems-A, 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.  Google Scholar

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R. J. DuanA. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Eqquations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.  Google Scholar

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M. D. FrancescoA. Lorz and P. Markowich, hemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Continous Dynam. Systems-A, 28 (2010), 1437-1453.  doi: 10.3934/dcds.2010.28.1437.  Google Scholar

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M. HenryD. Hilhorst and R. Schatzle, Convergence to a viscosity solution for an advection-reaction-diffusion equation arising from a chemotaxis-growth model, Hiroshima Math. J., 29 (1999), 591-630.  doi: 10.32917/hmj/1206124856.  Google Scholar

[6]

M. A. HerreroE. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  doi: 10.1088/0951-7715/10/6/016.  Google Scholar

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M. A.Herrero and J.J. L.Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.   Google Scholar

[8]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.   Google Scholar

[9]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.   Google Scholar

[10]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[11]

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.  Google Scholar

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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.  Google Scholar

[13]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.  Google Scholar

[14]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  Google Scholar

[15]

C. Miao, J. Wu and Z.Zhang, Littlewood-Paley Theory and Applications to Fluid Dynamics Equations, Monogr. Modern Pure Math. Science Press, Beijing, 42, 2012. Google Scholar

[16]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth., Physica A, 230 (1996), 499-543.   Google Scholar

[17]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[18]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phy. D, 240 (2011), 363-375.   Google Scholar

[19]

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), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[20]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Continous Dynam. Systems-A, 32 (2012), 1901-1914.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[21]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[22]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Part. Differ. Eq., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[23]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. KesskerR. E. Goldstein and H. L. Swinney, Bacterial swimming and oxygen transport near constant lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2285.   Google Scholar

[24]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[25]

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.  Google Scholar

[26]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[27]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., textbf348 (2008), 708–729. doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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.  Google Scholar

[31]

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.  Google Scholar

[32]

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.  Google Scholar

[33]

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.  Google Scholar

[34]

Q. Zhang, On the inviscid limit of the three dimensional incompressible chemotaxis-Navier-Stokes equations, Nonlinear Anal. Real World Appl., 27 (2016), 70-79.  doi: 10.1016/j.nonrwa.2015.07.008.  Google Scholar

[35]

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.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled Chemotaxis-fluid equations, Discrete Continous Dynam. Systems-A, 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[3]

R. J. DuanA. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Eqquations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.  Google Scholar

[4]

M. D. FrancescoA. Lorz and P. Markowich, hemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Continous Dynam. Systems-A, 28 (2010), 1437-1453.  doi: 10.3934/dcds.2010.28.1437.  Google Scholar

[5]

M. HenryD. Hilhorst and R. Schatzle, Convergence to a viscosity solution for an advection-reaction-diffusion equation arising from a chemotaxis-growth model, Hiroshima Math. J., 29 (1999), 591-630.  doi: 10.32917/hmj/1206124856.  Google Scholar

[6]

M. A. HerreroE. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  doi: 10.1088/0951-7715/10/6/016.  Google Scholar

[7]

M. A.Herrero and J.J. L.Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.   Google Scholar

[8]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.   Google Scholar

[9]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.   Google Scholar

[10]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.  Google Scholar

[14]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  Google Scholar

[15]

C. Miao, J. Wu and Z.Zhang, Littlewood-Paley Theory and Applications to Fluid Dynamics Equations, Monogr. Modern Pure Math. Science Press, Beijing, 42, 2012. Google Scholar

[16]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth., Physica A, 230 (1996), 499-543.   Google Scholar

[17]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[18]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phy. D, 240 (2011), 363-375.   Google Scholar

[19]

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), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[20]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Continous Dynam. Systems-A, 32 (2012), 1901-1914.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[21]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[22]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Part. Differ. Eq., 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[23]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. KesskerR. E. Goldstein and H. L. Swinney, Bacterial swimming and oxygen transport near constant lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2285.   Google Scholar

[24]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[25]

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.  Google Scholar

[26]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[27]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., textbf348 (2008), 708–729. doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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.  Google Scholar

[31]

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.  Google Scholar

[32]

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.  Google Scholar

[33]

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.  Google Scholar

[34]

Q. Zhang, On the inviscid limit of the three dimensional incompressible chemotaxis-Navier-Stokes equations, Nonlinear Anal. Real World Appl., 27 (2016), 70-79.  doi: 10.1016/j.nonrwa.2015.07.008.  Google Scholar

[35]

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.  Google Scholar

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