doi: 10.3934/dcdsb.2022088
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On the fractional chemotaxis Navier-Stokes system in the critical spaces

1. 

Departamento de Matemática, Universidade de Pernambuco, Nazaré da Mata-PE, Brazil

2. 

Departamento de Matemática, Universidade Federal de Pernambuco, Recife-PE, Brazil

3. 

Departamento de Matemática, Universidade Federal Rural de Pernambuco, Recife-PE, Brazil

*Corresponding author: Claudio Cuevas

Received  October 2021 Revised  March 2022 Early access May 2022

We consider the fractional chemotaxis Navier-Stokes equations which are the fractional Keller-Segel model coupled with the Navier-Stokes fluid in the whole space, and prove the existence of global mild solutions with the small critical initial data in Besov-Morrey spaces. Our results enable us to obtain the self-similar solutions provided the initial data are homogeneous functions with small norms and considering the case of chemical attractant without degradation rate. Moreover, we show the asymptotic stability of solutions as the time goes to infinity and obtain a class of asymptotically self-similar ones.

Citation: Joelma Azevedo, Claudio Cuevas, Jarbas Dantas, Clessius Silva. On the fractional chemotaxis Navier-Stokes system in the critical spaces. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022088
References:
[1]

J. AzevedoC. Cuevas and E. Henríquez, Existence and asymptotic behaviour for the time-fractional Keller-Segel model for chemotaxis, Math. Nach., 292 (2019), 462-480.  doi: 10.1002/mana.201700237.

[2]

P. BilerM. CannoneI. A. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann., 330 (2004), 693-708.  doi: 10.1007/s00208-004-0565-7.

[3]

M. Braukhoff, Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1013-1039.  doi: 10.1016/j.anihpc.2016.08.003.

[4]

S. Campanato, Proprietà di uma famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa, 18 (1964) 137–160.

[5]

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.

[6]

H. J. Choe and B. Lkhagvasuren, Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, J. Math. Anal. Appl., 446 (2017), 1415-1426.  doi: 10.1016/j.jmaa.2016.09.050.

[7]

L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in $\mathbb{R}^d$, C. R. Math. Acad. Sci. Paris, 342 (2006), 745–750. doi: 10.1016/j.crma.2006.03.008.

[8]

P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.

[9]

C. DombrowskiL. CisnerosS. ChatkaewR. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103.  doi: 10.1103/PhysRevLett.93.098103.

[10]

R. DuanA. 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.

[11]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.  doi: 10.1093/imrn/rns270.

[12]

L. C. F. Ferreira and M. Postigo, Global well-posedness and asymptotic behavior in Besov-Morrey spaces for chemotaxis-Navier-Stokes fluids, J. Math. Phys., 60 (2019), 061502.  doi: 10.1063/1.5080248.

[13]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with apllications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.

[14]

H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations, 19 (1994), 959–1014. doi: 10.1080/03605309408821042.

[15]

H. Kozono and M. Yamazaki, Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains, Nonlinear Anal., 38 (1999), 959-970.  doi: 10.1016/S0362-546X(98)00145-X.

[16]

H. KozonoM. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270 (2016), 1663-1683.  doi: 10.1016/j.jfa.2015.10.016.

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd ed. New York: Science Publishers, 1969.

[18]

T. A. M. Langlands and B. I. Henry, Fractional chemotaxis diffusion equations, Phys. Rev. E, 81 (2010), 051102, 12 pp. doi: 10.1103/PhysRevE.81.051102.

[19]

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.

[20]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, 2002. doi: 10.1201/9781420035674.

[21]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I, Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, Vol. 3, New York: Oxford University Press, 1996.

[22]

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.

[23]

A. L. Mazzucato, Besov-Morrey spaces: Function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.

[24]

T. Nagai and T. Yamada, Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space, J. Math. Anal. Appl., 336 (2007), 704-726.  doi: 10.1016/j.jmaa.2007.03.014.

[25]

J. Peetre, On the theory of $\mathcal{L}_{p, \lambda}$ spaces, J. Funct. Anal., 4 (1969), 71-87.  doi: 10.1016/0022-1236(69)90022-6.

[26]

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.

[27]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.

[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, 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.

[30]

A. Yagi, Norm behavior of solutions to the parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265. 

[31]

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.

[32]

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.

[33]

J. Zhao and J. Zhou, Temporal decay in negative Besov spaces for the 3D coupled chemotaxis-fluid equations, Nonlinear Anal. Real World Appl., 42 (2018), 160-179.  doi: 10.1016/j.nonrwa.2018.01.001.

show all references

References:
[1]

J. AzevedoC. Cuevas and E. Henríquez, Existence and asymptotic behaviour for the time-fractional Keller-Segel model for chemotaxis, Math. Nach., 292 (2019), 462-480.  doi: 10.1002/mana.201700237.

[2]

P. BilerM. CannoneI. A. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann., 330 (2004), 693-708.  doi: 10.1007/s00208-004-0565-7.

[3]

M. Braukhoff, Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1013-1039.  doi: 10.1016/j.anihpc.2016.08.003.

[4]

S. Campanato, Proprietà di uma famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa, 18 (1964) 137–160.

[5]

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.

[6]

H. J. Choe and B. Lkhagvasuren, Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, J. Math. Anal. Appl., 446 (2017), 1415-1426.  doi: 10.1016/j.jmaa.2016.09.050.

[7]

L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in $\mathbb{R}^d$, C. R. Math. Acad. Sci. Paris, 342 (2006), 745–750. doi: 10.1016/j.crma.2006.03.008.

[8]

P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.

[9]

C. DombrowskiL. CisnerosS. ChatkaewR. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103.  doi: 10.1103/PhysRevLett.93.098103.

[10]

R. DuanA. 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.

[11]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.  doi: 10.1093/imrn/rns270.

[12]

L. C. F. Ferreira and M. Postigo, Global well-posedness and asymptotic behavior in Besov-Morrey spaces for chemotaxis-Navier-Stokes fluids, J. Math. Phys., 60 (2019), 061502.  doi: 10.1063/1.5080248.

[13]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with apllications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.

[14]

H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations, 19 (1994), 959–1014. doi: 10.1080/03605309408821042.

[15]

H. Kozono and M. Yamazaki, Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains, Nonlinear Anal., 38 (1999), 959-970.  doi: 10.1016/S0362-546X(98)00145-X.

[16]

H. KozonoM. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270 (2016), 1663-1683.  doi: 10.1016/j.jfa.2015.10.016.

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd ed. New York: Science Publishers, 1969.

[18]

T. A. M. Langlands and B. I. Henry, Fractional chemotaxis diffusion equations, Phys. Rev. E, 81 (2010), 051102, 12 pp. doi: 10.1103/PhysRevE.81.051102.

[19]

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.

[20]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, 2002. doi: 10.1201/9781420035674.

[21]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I, Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, Vol. 3, New York: Oxford University Press, 1996.

[22]

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.

[23]

A. L. Mazzucato, Besov-Morrey spaces: Function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.

[24]

T. Nagai and T. Yamada, Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space, J. Math. Anal. Appl., 336 (2007), 704-726.  doi: 10.1016/j.jmaa.2007.03.014.

[25]

J. Peetre, On the theory of $\mathcal{L}_{p, \lambda}$ spaces, J. Funct. Anal., 4 (1969), 71-87.  doi: 10.1016/0022-1236(69)90022-6.

[26]

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.

[27]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.

[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, 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.

[30]

A. Yagi, Norm behavior of solutions to the parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265. 

[31]

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.

[32]

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.

[33]

J. Zhao and J. Zhou, Temporal decay in negative Besov spaces for the 3D coupled chemotaxis-fluid equations, Nonlinear Anal. Real World Appl., 42 (2018), 160-179.  doi: 10.1016/j.nonrwa.2018.01.001.

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