May  2020, 19(5): 2713-2735. doi: 10.3934/cpaa.2020118

Low modes regularity criterion for a chemotaxis-Navier-Stokes system

Department of Mathematics, Stat. and Comp. Sci., University of Illinois at Chicago, Chicago, IL 60607, USA

* Corresponding author

Received  April 2019 Revised  October 2019 Published  March 2020

Fund Project: The authors were partially supported by NSF grant DMS–1815069

In this paper we study the regularity problem of a three dimensional chemotaxis-Navier-Stokes system. A new regularity criterion in terms of only low modes of the oxygen concentration and the fluid velocity is obtained via a wavenumber splitting approach. The result improves certain existing criteria in the literature.

Citation: Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118
References:
[1]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehrender Mathematischen Wissenschaften, Vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

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A. CheskidovM. Dai and L. Kavlie, Determining modes for the 3D Navier-Stokes equations, Physica D, 374/375 (2018), 1-9.  doi: 10.1016/j.physd.2017.11.014.  Google Scholar

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A. Cheskidov and M. Dai, Regularity criteria for the 3D Navier-Stokes and MHD equations, Proc. Edinb. Math. Soc., (2019), To appear. doi: 10.1016/j.physd.2017.11.014.  Google Scholar

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A. Cheskidov and R. Shvydkoy, A unified approach to regularity problems for the 3D Navier-Stokes and Euler equations: the use of Kolmogorov's dissipation range, J. Math. Fluid Mech., 16 (2014), 263-273.  doi: 10.1007/s00021-014-0167-4.  Google Scholar

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M. Dai, Regularity problem for the nematic LCD system with Q-tensor in $\mathbb R^3$, SIAM J. Math. Anal., 49 (2017), 5007-5030.  doi: 10.1137/16M109137X.  Google Scholar

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C. DombrowskiL. CisnerosS. ChatkaewR. Goldstein and J. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93.098103 (2004), 1-4.  doi: 10.1103/PhysRevLett.93.098103.  Google Scholar

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

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L. EskauriazaG. A. Serëgin and V. Šverak, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar

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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.  Google Scholar

[15]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

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J. JiangH. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Anal., 92 (2015), 249-258.  doi: 10.3233/ASY-141276.  Google Scholar

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J. JiangH. Wu and S. Zheng, Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant, J. Differ. Equ., 264 (2018), 5432-5464.  doi: 10.1016/j.jde.2018.01.004.  Google Scholar

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E. Keller and L. Segel, Travelling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

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H. KozonoT. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z., 242 (2002), 251-278.  doi: 10.1007/s002090100332.  Google Scholar

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N. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, Vol. 96, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.  Google Scholar

[21]

O. A. Ladyzhenskaya, Uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, English transl., Sem. Math. V. A. Steklov Math. Inst. Leningrad, 5 (1969), 60-66.   Google Scholar

[22]

O. Ladyzhenskaya, V. A. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[23]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC Research Notes in Mathematics, Vol. 431, Chapman and Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[24]

J. Liu and A. Lorz, A coupled chemotaxis fluid model: global existence, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 28 (2011), 643-652.  doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[25]

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

[26]

F. Planchon, An extension of the Beale-Kato-Majda criterion for the Euler Equations, Commun. Math. Phys., 232 (2003), 319-326.  doi: 10.1007/s00220-002-0744-x.  Google Scholar

[27]

G. Prodi, Teorema di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.  doi: 10.1007/BF02410664.  Google Scholar

[28]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, Birkhäuser, Basel, 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[29]

G. Rosen, Steady-state distribution of bacteria chemotactic towards oxygen, Bull. Math. Biol., 40 (1978), 641-674.  doi: 10.1016/S0092-8240(78)80025-1.  Google Scholar

[30]

J. Serrin, The initial value problem for the Navier-Stokes equations, in Nonlinear Problems (Proc. Sympos., Madison, Wis.), Univ. of Wisconsin Press, Madison, Wis, (1963), 69–98.  Google Scholar

[31]

I. TuvalL. CisnerosC. DombrowskiC. WolgemuthJ. Kessler and R. 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.  Google Scholar

[32]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[33]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differ. Equ., 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[34]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehrender Mathematischen Wissenschaften, Vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

J. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[3]

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.  Google Scholar

[4]

M. ChaeK. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Commun. Partial Differ. Equ., 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.  Google Scholar

[5]

A. Cheskidov and M. Dai, Determining modes for the surface quasi-geostrophic equation, Physica D, 376/377 (2018), 204-215.  doi: 10.1016/j.physd.2018.03.003.  Google Scholar

[6]

A. CheskidovM. Dai and L. Kavlie, Determining modes for the 3D Navier-Stokes equations, Physica D, 374/375 (2018), 1-9.  doi: 10.1016/j.physd.2017.11.014.  Google Scholar

[7]

A. Cheskidov and M. Dai, Regularity criteria for the 3D Navier-Stokes and MHD equations, Proc. Edinb. Math. Soc., (2019), To appear. doi: 10.1016/j.physd.2017.11.014.  Google Scholar

[8]

A. Cheskidov and R. Shvydkoy, A unified approach to regularity problems for the 3D Navier-Stokes and Euler equations: the use of Kolmogorov's dissipation range, J. Math. Fluid Mech., 16 (2014), 263-273.  doi: 10.1007/s00021-014-0167-4.  Google Scholar

[9]

M. Dai, Regularity problem for the nematic LCD system with Q-tensor in $\mathbb R^3$, SIAM J. Math. Anal., 49 (2017), 5007-5030.  doi: 10.1137/16M109137X.  Google Scholar

[10]

C. DombrowskiL. CisnerosS. ChatkaewR. Goldstein and J. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93.098103 (2004), 1-4.  doi: 10.1103/PhysRevLett.93.098103.  Google Scholar

[11]

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

[12]

L. EskauriazaG. A. Serëgin and V. Šverak, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.  Google Scholar

[13]

L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics, Vol. 250, 2$^{nd}$ edition, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.  Google Scholar

[14]

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.  Google Scholar

[15]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[16]

J. JiangH. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Anal., 92 (2015), 249-258.  doi: 10.3233/ASY-141276.  Google Scholar

[17]

J. JiangH. Wu and S. Zheng, Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant, J. Differ. Equ., 264 (2018), 5432-5464.  doi: 10.1016/j.jde.2018.01.004.  Google Scholar

[18]

E. Keller and L. Segel, Travelling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[19]

H. KozonoT. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z., 242 (2002), 251-278.  doi: 10.1007/s002090100332.  Google Scholar

[20]

N. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, Vol. 96, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.  Google Scholar

[21]

O. A. Ladyzhenskaya, Uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, English transl., Sem. Math. V. A. Steklov Math. Inst. Leningrad, 5 (1969), 60-66.   Google Scholar

[22]

O. Ladyzhenskaya, V. A. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[23]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC Research Notes in Mathematics, Vol. 431, Chapman and Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[24]

J. Liu and A. Lorz, A coupled chemotaxis fluid model: global existence, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 28 (2011), 643-652.  doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[25]

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

[26]

F. Planchon, An extension of the Beale-Kato-Majda criterion for the Euler Equations, Commun. Math. Phys., 232 (2003), 319-326.  doi: 10.1007/s00220-002-0744-x.  Google Scholar

[27]

G. Prodi, Teorema di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.  doi: 10.1007/BF02410664.  Google Scholar

[28]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, Birkhäuser, Basel, 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[29]

G. Rosen, Steady-state distribution of bacteria chemotactic towards oxygen, Bull. Math. Biol., 40 (1978), 641-674.  doi: 10.1016/S0092-8240(78)80025-1.  Google Scholar

[30]

J. Serrin, The initial value problem for the Navier-Stokes equations, in Nonlinear Problems (Proc. Sympos., Madison, Wis.), Univ. of Wisconsin Press, Madison, Wis, (1963), 69–98.  Google Scholar

[31]

I. TuvalL. CisnerosC. DombrowskiC. WolgemuthJ. Kessler and R. 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.  Google Scholar

[32]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[33]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differ. Equ., 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[34]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

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