June  2021, 26(6): 3409-3425. doi: 10.3934/dcdsb.2020237

Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces

1. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

2. 

ICT School, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, South Korea

3. 

School of Mathematical Sciences, Qufu Normal University, Qufu, 273100, China

4. 

Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330032, China

*Corresponding author: Zunwei Fu

Received  March 2020 Revised  May 2020 Published  June 2021 Early access  August 2020

Fund Project: This paper was partially supported by the National Natural Science Foundation of China (Grant No. 11801236), the Postdoctoral Science Foundation of China (Grant Nos. 2018M632593, 2019M660555), the Natural Science Foundation of Gansu Province for Young Scholars (Grant No. 18JR3RA102), the innovation capacity improvement project for colleges and universities of Gansu Province (Grant No. 2019A-011), the Natural Science Foundation of Jiangxi Province for Young Scholars (Grant No. 20181BAB211001), the Postdoctoral Science Foundation of Jiangxi Province (Grant No. 2017KY23) and Educational Commission Science Programm of Jiangxi Province (Grant No. GJJ190272)

The paper is concerned with the Navier-Stokes-Nernst-Planck-Poisson system arising from electrohydrodynamics in $ \mathbb{R}^d $. By means of the implicit function theorem, we prove the global existence of mild solutions for Cauchy problem of this system with small initial data in critical Besov-Morrey spaces. In comparison to the previous works, our existence result provides a new class of initial data, for which the problem is global solvability. Meanwhile, based on the so-called Gevrey estimates, we verify that the obtained mild solutions are analytic in the spatial variables. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Furthermore, decay estimates of higher-order derivatives of solutions are deduced in Morrey spaces.

Citation: Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237
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D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.  doi: 10.1215/S0012-7094-75-04265-9.

[2]

H. BaeA. Biswas and E. Tadmor, Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Ration. Mech. Anal., 205 (2012), 963-991.  doi: 10.1007/s00205-012-0532-5.

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H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

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M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506. doi: 10.1103/PhysRevE.70.021506.

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M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes(French), Séminaire sur les équations aux Dérivées Partielles, 1993–1994.

[6]

M. Cannone and G. Wu, Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal., 75 (2012) 3754–3760. doi: 10.1016/j.na.2012.01.029.

[7]

C. DengJ. Zhao and S. Cui, Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal., 73 (2010), 2088-2100.  doi: 10.1016/j.na.2010.05.037.

[8]

C. DengJ. Zhao and S. Cui, Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices, J. Math. Anal. Appl., 377 (2011), 392-405.  doi: 10.1016/j.jmaa.2010.11.011.

[9]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3.

[10]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.

[11]

C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, arXiv: 1310.2141.

[12]

T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type, J. Funct. Anal., 267 (2014), 1321-1337.  doi: 10.1016/j.jfa.2014.05.022.

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J. W. Joseph, Analytical approaches to charge transport in a moving medium, Transport Theory Statist. Phys., 31 (2002), 333-366.  doi: 10.1081/TT-120015505.

[14]

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

[15]

T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat.(N.S.), 22 (1992), 127–155. doi: 10.1007/BF01232939.

[16]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.

[17]

P. Konieczny and T. Yoneda, On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations, J. Differential Equations, 250 (2011), 3859-3873.  doi: 10.1016/j.jde.2011.01.003.

[18]

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.

[19]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304. doi: 10.1002/cpa.20361.

[20] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.
[21]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[22]

Q. Liu and S. Cui, Regularizing rate estimates for mild solutions of the incompressible magneto-hydrodynamic system, Commun. Pure Appl. Anal., 11 (2012), 1643–1660. doi: 10.3934/cpaa.2012.11.1643.

[23]

Q. LiuJ. Zhao and S. Cui, Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl., 191 (2012), 293-309.  doi: 10.1007/s10231-010-0184-8.

[24]

F. Li, Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differential Equations, 246 (2009), 3620-3641.  doi: 10.1016/j.jde.2009.01.027.

[25]

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.

[26]

J. Newman and K. Thomas-Alyea, Electrochemical Systems(3rd Edition), J. Wiley, Hoboken, 2004.

[27]

M. Oliver and E. S. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in $\mathbb{R}^n$, J. Funct. Anal., 172 (2000), 1-18.  doi: 10.1006/jfan.1999.3550.

[28]

R. J. Ryham, An energetic variational approach to mathematical modeling of charged fluids: Charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, 83pp.

[29]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014.  doi: 10.1142/S0218202509003693.

[30]

J. Sun, M. Yang and S. Cui, Existence and analyticity of mild solutions for the 3D rotating Navier-Stokes equations, Ann. Mat. Pura Appl., 196 (2017), no. 4, 1203–1229. doi: 10.1007/s10231-016-0613-4.

[31]

M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407–1456. doi: 10.1080/03605309208820892.

[32]

M. Yang, Z. Fu and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math., 60 (2017), 1837-1856. doi: 10.1007/s11425-016-0490-y.

[33]

M. Yang and J. Sun, Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces, Commun. Pure Appl. Anal., 16 (2017), 1617-1639.  doi: 10.3934/cpaa.2017078.

[34]

M. YangZ. Fu and J. Sun, Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces, J. Differential Equations, 266 (2019), 5867-5894.  doi: 10.1016/j.jde.2018.10.050.

[35]

J. Zhao, C. Deng and S. Cui, Global well-posedness of a dissipative system arising in electrohydrodynamics in negative-order Besov spaces, J. Math. Phys., 51 (2010), 093101. doi: 10.1063/1.3484184.

[36]

J. ZhaoC. Deng and S. Cui, Well-posedness of a dissipative system modeling electrohydrodynamics in Lebesgue spaces, Differential Equations Appl., 3 (2011), 427-448.  doi: 10.7153/dea-03-27.

[37]

J. ZhaoQ. Liu and S. Cui, Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye-Hückel system, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 1-18.  doi: 10.1007/s00030-011-0115-4.

show all references

References:
[1]

D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.  doi: 10.1215/S0012-7094-75-04265-9.

[2]

H. BaeA. Biswas and E. Tadmor, Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Ration. Mech. Anal., 205 (2012), 963-991.  doi: 10.1007/s00205-012-0532-5.

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[4]

M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506. doi: 10.1103/PhysRevE.70.021506.

[5]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes(French), Séminaire sur les équations aux Dérivées Partielles, 1993–1994.

[6]

M. Cannone and G. Wu, Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal., 75 (2012) 3754–3760. doi: 10.1016/j.na.2012.01.029.

[7]

C. DengJ. Zhao and S. Cui, Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal., 73 (2010), 2088-2100.  doi: 10.1016/j.na.2010.05.037.

[8]

C. DengJ. Zhao and S. Cui, Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices, J. Math. Anal. Appl., 377 (2011), 392-405.  doi: 10.1016/j.jmaa.2010.11.011.

[9]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3.

[10]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.

[11]

C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, arXiv: 1310.2141.

[12]

T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type, J. Funct. Anal., 267 (2014), 1321-1337.  doi: 10.1016/j.jfa.2014.05.022.

[13]

J. W. Joseph, Analytical approaches to charge transport in a moving medium, Transport Theory Statist. Phys., 31 (2002), 333-366.  doi: 10.1081/TT-120015505.

[14]

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

[15]

T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat.(N.S.), 22 (1992), 127–155. doi: 10.1007/BF01232939.

[16]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.

[17]

P. Konieczny and T. Yoneda, On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations, J. Differential Equations, 250 (2011), 3859-3873.  doi: 10.1016/j.jde.2011.01.003.

[18]

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.

[19]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304. doi: 10.1002/cpa.20361.

[20] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.
[21]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[22]

Q. Liu and S. Cui, Regularizing rate estimates for mild solutions of the incompressible magneto-hydrodynamic system, Commun. Pure Appl. Anal., 11 (2012), 1643–1660. doi: 10.3934/cpaa.2012.11.1643.

[23]

Q. LiuJ. Zhao and S. Cui, Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl., 191 (2012), 293-309.  doi: 10.1007/s10231-010-0184-8.

[24]

F. Li, Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differential Equations, 246 (2009), 3620-3641.  doi: 10.1016/j.jde.2009.01.027.

[25]

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.

[26]

J. Newman and K. Thomas-Alyea, Electrochemical Systems(3rd Edition), J. Wiley, Hoboken, 2004.

[27]

M. Oliver and E. S. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in $\mathbb{R}^n$, J. Funct. Anal., 172 (2000), 1-18.  doi: 10.1006/jfan.1999.3550.

[28]

R. J. Ryham, An energetic variational approach to mathematical modeling of charged fluids: Charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, 83pp.

[29]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014.  doi: 10.1142/S0218202509003693.

[30]

J. Sun, M. Yang and S. Cui, Existence and analyticity of mild solutions for the 3D rotating Navier-Stokes equations, Ann. Mat. Pura Appl., 196 (2017), no. 4, 1203–1229. doi: 10.1007/s10231-016-0613-4.

[31]

M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407–1456. doi: 10.1080/03605309208820892.

[32]

M. Yang, Z. Fu and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math., 60 (2017), 1837-1856. doi: 10.1007/s11425-016-0490-y.

[33]

M. Yang and J. Sun, Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces, Commun. Pure Appl. Anal., 16 (2017), 1617-1639.  doi: 10.3934/cpaa.2017078.

[34]

M. YangZ. Fu and J. Sun, Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces, J. Differential Equations, 266 (2019), 5867-5894.  doi: 10.1016/j.jde.2018.10.050.

[35]

J. Zhao, C. Deng and S. Cui, Global well-posedness of a dissipative system arising in electrohydrodynamics in negative-order Besov spaces, J. Math. Phys., 51 (2010), 093101. doi: 10.1063/1.3484184.

[36]

J. ZhaoC. Deng and S. Cui, Well-posedness of a dissipative system modeling electrohydrodynamics in Lebesgue spaces, Differential Equations Appl., 3 (2011), 427-448.  doi: 10.7153/dea-03-27.

[37]

J. ZhaoQ. Liu and S. Cui, Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye-Hückel system, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 1-18.  doi: 10.1007/s00030-011-0115-4.

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