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September  2017, 16(5): 1617-1639. doi: 10.3934/cpaa.2017078

## Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces

 1 School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330032, China 2 Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author

Received  August 2016 Revised  March 2017 Published  May 2017

The paper deals with the Cauchy problem of Navier-Stokes-Nernst-Planck-Poisson system (NSNPP). First of all, based on so-called Gevrey regularity estimates, which is motivated by the works of Foias and Temam [J. Funct. Anal., 87 (1989), 359-369], we prove that the solutions are analytic in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly obtain higher-order derivatives of solutions in Besov and Lebesgue spaces. Finally, we prove that there exists a positive constant
 $\mathbb{C}$
such that if the initial data
 $(u_{0}, n_{0}, c_{0})=(u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$
satisfies
 \begin{aligned}&\|(n_{0}, c_{0},u_{0}^{h})\|_{\dot{B}^{-2+3/q}_{q, 1}× \dot{B}^{-2+3/q}_{q, 1}×\dot{B}^{-1+3/p}_{p, 1}}+\|u_{0}^{h}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{α}\|u_{0}^{3}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{1-α}≤q1/\mathbb{C}\end{aligned}
for
 $p, q, α$
with
 $1\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$
, then global existence of solutions with large initial vertical velocity component is established.
Citation: Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078
##### References:
 [1] A. Biswas, V. Martinez and P. Silva, On Gevrey regularity of the supercritical SQG equation in critical Besov spaces, J. Funct. Anal., 269 (2015), 3083-3119. doi: 10.1016/j.jfa.2015.08.010. Google Scholar [2] H. Bae, Existence and analyticity of Lei-Lin Solution to the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892. doi: 10.1090/S0002-9939-2015-12266-6. Google Scholar [3] 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. Google Scholar [4] A. Biswas, Gevrey regularity for a class of dissipative equations with applications to decay, J. Differ. Equ., 253 (2012), 2739-2764. doi: 10.1016/j.jde.2012.08.003. Google Scholar [5] A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3D Navier-Stokes equations with weighted $\ell^{p}$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188. doi: 10.1512/iumj.2007.56.2891. Google Scholar [6] J. Y. Chemin, Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50. doi: 10.1007/BF02791256. Google Scholar [7] J. Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable diffusion, J. Differ. Equ., 256 (2014), 223-252. doi: 10.1016/j.jde.2013.09.004. Google Scholar [8] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differ. Equ., 19 (1994), 959-1014. doi: 10.1080/03605309408821042. Google Scholar [9] M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, 3 (2004), 161-244. Google Scholar [10] R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November (2005).Google Scholar [11] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differ. Equ., 26 (2001), 1183-31233. doi: 10.1081/PDE-100106132. Google Scholar [12] C. Deng, J. Zhao and S. Cui, Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal. Theory Methods Appl., 73 (2010), 2088-2100. doi: 10.1016/j.na.2010.05.037. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. of Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [16] G. Gui and P. Zhang, Stability to the global large solutions of 3D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284. doi: 10.1016/j.aim.2010.03.022. Google Scholar [17] B. Hajer, Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar [18] J. Huang, M. Paicu and P. Zhang, Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209 (2013), 631-382. doi: 10.1007/s00205-013-0624-x. Google Scholar [19] 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. Google Scholar [20] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937. Google Scholar [21] Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math. , 64, 1297-1304. doi: 10.1002/cpa.20361. Google Scholar [22] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar [23] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. Google Scholar [24] M. Paicu, équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 225 (2010), 1248-1284. doi: 10.4171/RMI/420. Google Scholar [25] M. Paicu and P. Zhang, Global solutions to the 3D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759. doi: 10.1007/s00220-011-1350-6. Google Scholar [26] M. Paicu and P. Zhang, Global solutions to the 3D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584. doi: 10.1016/j.jfa.2012.01.022. Google Scholar [27] F. Planchon, Sur un in$\acute{e}$galit$\acute{e}$ de type Poincar$\acute{e}$, C. R. Acad. Sci. Paris S$\acute{e}$r. Ⅰ Math., 330 (2000), 21-23. doi: 10.1016/S0764-4442(00)88138-0. Google Scholar [28] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Methods for Nonlinear Evolution Equations, World Scientific, 2011. doi: 10.1142/9789814360746. Google Scholar [29] J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dyn. Partial Differ. Equ., 4 (2007), 227-245. doi: 10.4310/DPDE.2007.v4.n3.a2. Google Scholar [30] M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E., 70 (2004), 021506. Google Scholar [31] 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. Google Scholar [32] F. Li, Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differ. Equ., 246 (2009), 3620-3641. doi: 10.1016/j.jde.2009.01.027. Google Scholar [33] F. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919. doi: 10.1002/cpa.21402. Google Scholar [34] J. Newman and K. Thomas, Electrochemical Systems, thirded., John Wiley Sons, 2004. Google Scholar [35] R. Ryham, An energetic variational approach to mathematical modeling of charged fluids: charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, p. 83.Google Scholar [36] M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014. doi: 10.1142/S0218202509003693. Google Scholar [37] C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, http://arxiv.org/abs/1310.2141.Google Scholar [38] J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited, Dyn. Partial Differ. Equ., 11 (2014), 167-181. doi: 10.4310/DPDE.2014.v11.n2.a3. Google Scholar [39] C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity J. Math. Phys. , 56 091512 (2015). doi: 10.1063/1.4931467. Google Scholar [40] 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), 093-101. doi: 10.1063/1.3484184. Google Scholar [41] 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. Google Scholar [42] J. Zhao, T. Zhang and Q Liu, Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space, Discrete Contin. Dyn. Syst., 35 (2015), 555-582. doi: 10.3934/dcds.2015.35.555. Google Scholar

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##### References:
 [1] A. Biswas, V. Martinez and P. Silva, On Gevrey regularity of the supercritical SQG equation in critical Besov spaces, J. Funct. Anal., 269 (2015), 3083-3119. doi: 10.1016/j.jfa.2015.08.010. Google Scholar [2] H. Bae, Existence and analyticity of Lei-Lin Solution to the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892. doi: 10.1090/S0002-9939-2015-12266-6. Google Scholar [3] 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. Google Scholar [4] A. Biswas, Gevrey regularity for a class of dissipative equations with applications to decay, J. Differ. Equ., 253 (2012), 2739-2764. doi: 10.1016/j.jde.2012.08.003. Google Scholar [5] A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3D Navier-Stokes equations with weighted $\ell^{p}$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188. doi: 10.1512/iumj.2007.56.2891. Google Scholar [6] J. Y. Chemin, Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50. doi: 10.1007/BF02791256. Google Scholar [7] J. Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable diffusion, J. Differ. Equ., 256 (2014), 223-252. doi: 10.1016/j.jde.2013.09.004. Google Scholar [8] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differ. Equ., 19 (1994), 959-1014. doi: 10.1080/03605309408821042. Google Scholar [9] M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, 3 (2004), 161-244. Google Scholar [10] R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November (2005).Google Scholar [11] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differ. Equ., 26 (2001), 1183-31233. doi: 10.1081/PDE-100106132. Google Scholar [12] C. Deng, J. Zhao and S. Cui, Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal. Theory Methods Appl., 73 (2010), 2088-2100. doi: 10.1016/j.na.2010.05.037. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. of Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [16] G. Gui and P. Zhang, Stability to the global large solutions of 3D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284. doi: 10.1016/j.aim.2010.03.022. Google Scholar [17] B. Hajer, Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar [18] J. Huang, M. Paicu and P. Zhang, Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209 (2013), 631-382. doi: 10.1007/s00205-013-0624-x. Google Scholar [19] 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. Google Scholar [20] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937. Google Scholar [21] Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math. , 64, 1297-1304. doi: 10.1002/cpa.20361. Google Scholar [22] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar [23] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. Google Scholar [24] M. Paicu, équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 225 (2010), 1248-1284. doi: 10.4171/RMI/420. Google Scholar [25] M. Paicu and P. Zhang, Global solutions to the 3D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759. doi: 10.1007/s00220-011-1350-6. Google Scholar [26] M. Paicu and P. Zhang, Global solutions to the 3D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584. doi: 10.1016/j.jfa.2012.01.022. Google Scholar [27] F. Planchon, Sur un in$\acute{e}$galit$\acute{e}$ de type Poincar$\acute{e}$, C. R. Acad. Sci. Paris S$\acute{e}$r. Ⅰ Math., 330 (2000), 21-23. doi: 10.1016/S0764-4442(00)88138-0. Google Scholar [28] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Methods for Nonlinear Evolution Equations, World Scientific, 2011. doi: 10.1142/9789814360746. Google Scholar [29] J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dyn. Partial Differ. Equ., 4 (2007), 227-245. doi: 10.4310/DPDE.2007.v4.n3.a2. Google Scholar [30] M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E., 70 (2004), 021506. Google Scholar [31] 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. Google Scholar [32] F. Li, Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differ. Equ., 246 (2009), 3620-3641. doi: 10.1016/j.jde.2009.01.027. Google Scholar [33] F. Lin, Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919. doi: 10.1002/cpa.21402. Google Scholar [34] J. Newman and K. Thomas, Electrochemical Systems, thirded., John Wiley Sons, 2004. Google Scholar [35] R. Ryham, An energetic variational approach to mathematical modeling of charged fluids: charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, p. 83.Google Scholar [36] M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014. doi: 10.1142/S0218202509003693. Google Scholar [37] C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, http://arxiv.org/abs/1310.2141.Google Scholar [38] J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited, Dyn. Partial Differ. Equ., 11 (2014), 167-181. doi: 10.4310/DPDE.2014.v11.n2.a3. Google Scholar [39] C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity J. Math. Phys. , 56 091512 (2015). doi: 10.1063/1.4931467. Google Scholar [40] 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), 093-101. doi: 10.1063/1.3484184. Google Scholar [41] 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. Google Scholar [42] J. Zhao, T. Zhang and Q Liu, Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space, Discrete Contin. Dyn. Syst., 35 (2015), 555-582. doi: 10.3934/dcds.2015.35.555. Google Scholar
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