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Weak time discretization for slow-fast stochastic reaction-diffusion equations
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 |
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.
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. Bae, A. 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. Deng, J. 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. Deng, J. 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. Google Scholar |
[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. Liu, J. 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. Google Scholar |
[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. Yang, Z. 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. Zhao, C. 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. Zhao, Q. 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. Bae, A. 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. Deng, J. 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. Deng, J. 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. Google Scholar |
[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. Liu, J. 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. Google Scholar |
[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. Yang, Z. 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. Zhao, C. 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. Zhao, Q. 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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