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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
$1<p<q≤ 2p<\infty, \frac{1}{p}+\frac{1}{q}>\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$
, then global existence of solutions with large initial vertical velocity component is established.
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. BiswasV. 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. 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.  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. CheminM. 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. DengJ. 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. 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.  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. HajerY. 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. HuangM. 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. WangZ. HuoC. 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. BazantK. 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. ZhaoC. 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. 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.  Google Scholar

[42]

J. ZhaoT. 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. BiswasV. 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. 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.  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. CheminM. 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. DengJ. 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. 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.  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. HajerY. 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. HuangM. 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. WangZ. HuoC. 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. BazantK. 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. ZhaoC. 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. 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.  Google Scholar

[42]

J. ZhaoT. 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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