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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

Minghua Yang 1, and  Jinyi Sun 2,,

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
$1<p<q≤ 2p<\infty, \frac{1}{p}+\frac{1}{q}>\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$
, then global existence of solutions with large initial vertical velocity component is established.
Keywords: Nernst-Planck-Poisson system, Navier-Stokes system, Gevrey regularity, global solutions, Besov spaces.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 35E15.
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

show all references

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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