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January  2015, 35(1): 555-582. doi: 10.3934/dcds.2015.35.555

Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space

Jihong Zhao 1, , Ting Zhang 2, and  Qiao Liu 3,

1. 

College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
2. 

Department of Mathematics, Zhejiang University, Hangzhou 310027
3. 

Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081

Received  November 2013 Revised  June 2014 Published  August 2014

In this paper, we are concerned with a model arising from electro-hydrodynamics, which is a coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. The local well-posedness and global well-posedness with small initial data to the 3-D Cauchy problem of this system are established in the critical Besov space $\dot{B}^{-1+\frac{3}{p}}_{p,1}(\mathbb{R}^{3})\times(\dot{B}^{-2+\frac{3}{q}}_{q,1}(\mathbb{R}^{3}))^{2}$ with suitable choices of $p, q$. Especially, we prove that there exist two positive constants $c_{0}, C_{0}$ depending on the coefficients of system except $\mu$ such that if \begin{equation*} \big(\|u_{0}^{h}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+(\mu+1)\|(v_{0},w_{0})\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{\frac{C_{0}}{\mu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}^{2}+1)\Big\}\leq c_{0}\mu, \end{equation*} then the above local solution can be extended to the global one. This result implies the global well-posedness of this system with large initial vertical velocity component.
Keywords: electro-hydrodynamics, Navier-Stokes equations, well-posedness, Poisson-Nernst-Planck equations, Besov space..
Mathematics Subject Classification: Primary: 35K15, 35K55; Secondary: 35Q35, 76A0.
Citation: Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555
References:
[1]

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

[2]

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J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. école Norm. Sup., 14 (1981), 209-246.  Google Scholar

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M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13 (1997), 515-541. doi: 10.4171/RMI/229.  Google Scholar

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M. Cannone, Y. Meyer and F. Planchon, Solutions Auto-similaires des ÉQuations de Navier-Stokes in $\mathbbR^3$, Exposé n. VIII, Séminaire X-EDP, Ecole Polytechnique, 1994.  Google Scholar

[6]

J.-Y. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50. doi: 10.1007/BF02791256.  Google Scholar

[7]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.  Google Scholar

[8]

J.-Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys., 272 (2007), 529-566. doi: 10.1007/s00220-007-0236-0.  Google Scholar

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R. Danchin, Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.  Google Scholar

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R. Danchin, Fourier Analysis Methods for PDE's, http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf, 2005. Google Scholar

[11]

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

[12]

E. T. Enikov and B. J. Nelson, Electrotransport and deformation model of ion exchange membrane based actuators, Smart Structures and Materials, 3978 (2000), 129-139. doi: 10.1117/12.387771.  Google Scholar

[13]

E. T. Enikov and G. S. Seo, Analysis of water and proton fluxes in ion-exchange polymer-metal composite (IPMC) actuators subjected to large external potentials, Sensors and Actuators, 122 (2005), 264-272. doi: 10.1016/j.sna.2005.02.042.  Google Scholar

[14]

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

[15]

Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[16]

J. Huang, M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system with rough density, Progress in Nonlinear Differential Equations and Their Applications, 84 (2013), 159-180. doi: 10.1007/978-1-4614-6348-1_9.  Google Scholar

[17]

J. W. Jerome, Analytical approaches to charge transport in a moving medium, Tran. Theo. Stat. Phys., 31 (2002), 333-366. doi: 10.1081/TT-120015505.  Google Scholar

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J. W. Jerome, The steady boundary value problem for charged incompressible fluids: PNP/Navier-Stokes systems, Nonlinear Anal., 74 (2011), 7486-7498. doi: 10.1016/j.na.2011.08.003.  Google Scholar

[19]

J. W. Jerome and R. Sacco, Global weak solutions for an incompressible charged fluid with multi-scale couplings: Initial-boundary-value problem, Nonlinear Anal., 71 (2009), e2487-e2497. doi: 10.1016/j.na.2009.05.047.  Google Scholar

[20]

T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in $\mathbbR^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[21]

T. Kato and G. Ponce, Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces $L_p^s(\mathbbR^2)$, Rev. Mat. Iberoam., 2 (1986), 73-88. doi: 10.4171/RMI/26.  Google Scholar

[22]

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

[23]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Research Notes in Mathematics, Chapman & Hall/CRC, 2002. doi: 10.1201/9781420035674.  Google Scholar

[24]

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

[25]

M. Longaretti, B. Chini, J. W. Jerome and R. Sacco, Electrochemical modeling and characterization of voltage operated channels in nano-bio-electronics, Sensor Letters, 6 (2008), 49-56. doi: 10.1166/sl.2008.010.  Google Scholar

[26]

M. Longaretti, B. Chini, J. W. Jerome and R. Sacco, Computational modeling and simulation of complex systems in bio-electronics, J. Computational Electronics, 7 (2008), 10-13. Google Scholar

[27]

M. Longaretti, G. Marino, B. Chini, J. W. Jerome and R. Sacco, Computational models in nano-bio-electronics: Simulation of ionic transport in voltage operated channels, J. Nanoscience and Nanotechnology, 8 (2008), 3686-3694. Google Scholar

[28]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Commun. Math. Phys., 307 (2011), 713-759. doi: 10.1007/s00220-011-1350-6.  Google Scholar

[29]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584. doi: 10.1016/j.jfa.2012.01.022.  Google Scholar

[30]

F. Planchon, Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 21-23. doi: 10.1016/S0764-4442(00)88138-0.  Google Scholar

[31]

I. Rubinstein, Electro-Diffusion of Ions, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611970814.  Google Scholar

[32]

R. J. Ryham, Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics,, , ().   Google Scholar

[33]

R. J. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 649-661. doi: 10.3934/dcdsb.2007.8.649.  Google Scholar

[34]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1015. doi: 10.1142/S0218202509003693.  Google Scholar

[35]

M. Shahinpoor and K. J. Kim, Ionic polymer-metal composites: III. Modeling and simulation as biomimetic sensors, actuators, transducers, and artificial muscles, Smart Mater. Struct., 13 (2004), 1362-1388. doi: 10.1088/0964-1726/13/6/009.  Google Scholar

[36]

H. Triebel, Theory of Function Spaces, Monogr. Math., 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[37]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Commun. Math. Phys., 287 (2009), 211-224. doi: 10.1007/s00220-008-0631-1.  Google Scholar

[38]

T. Zhang and D. Y. Fang, Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations, J. Math. Pures Appl., 90 (2008), 413-449. doi: 10.1016/j.matpur.2008.06.008.  Google Scholar

[39]

J. Zhao, C. Deng and S. Cui, Global well-posedness of a dissipative system arising in electrohydrodynamics in negative-order Besov spaces, J. Math. Physics, 51 (2010), 093101, 17 pp. doi: 10.1063/1.3484184.  Google Scholar

[40]

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

show all references

References:
[1]

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

[2]

P. B. Balbuena and Y. Wang, Lithium-ion Batteries, Solid-electrolyte Interphase, Imperial College Press, 2004. doi: 10.1142/p291.  Google Scholar

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. école Norm. Sup., 14 (1981), 209-246.  Google Scholar

[4]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13 (1997), 515-541. doi: 10.4171/RMI/229.  Google Scholar

[5]

M. Cannone, Y. Meyer and F. Planchon, Solutions Auto-similaires des ÉQuations de Navier-Stokes in $\mathbbR^3$, Exposé n. VIII, Séminaire X-EDP, Ecole Polytechnique, 1994.  Google Scholar

[6]

J.-Y. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50. doi: 10.1007/BF02791256.  Google Scholar

[7]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.  Google Scholar

[8]

J.-Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys., 272 (2007), 529-566. doi: 10.1007/s00220-007-0236-0.  Google Scholar

[9]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.  Google Scholar

[10]

R. Danchin, Fourier Analysis Methods for PDE's, http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf, 2005. Google Scholar

[11]

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

[12]

E. T. Enikov and B. J. Nelson, Electrotransport and deformation model of ion exchange membrane based actuators, Smart Structures and Materials, 3978 (2000), 129-139. doi: 10.1117/12.387771.  Google Scholar

[13]

E. T. Enikov and G. S. Seo, Analysis of water and proton fluxes in ion-exchange polymer-metal composite (IPMC) actuators subjected to large external potentials, Sensors and Actuators, 122 (2005), 264-272. doi: 10.1016/j.sna.2005.02.042.  Google Scholar

[14]

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

[15]

Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[16]

J. Huang, M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system with rough density, Progress in Nonlinear Differential Equations and Their Applications, 84 (2013), 159-180. doi: 10.1007/978-1-4614-6348-1_9.  Google Scholar

[17]

J. W. Jerome, Analytical approaches to charge transport in a moving medium, Tran. Theo. Stat. Phys., 31 (2002), 333-366. doi: 10.1081/TT-120015505.  Google Scholar

[18]

J. W. Jerome, The steady boundary value problem for charged incompressible fluids: PNP/Navier-Stokes systems, Nonlinear Anal., 74 (2011), 7486-7498. doi: 10.1016/j.na.2011.08.003.  Google Scholar

[19]

J. W. Jerome and R. Sacco, Global weak solutions for an incompressible charged fluid with multi-scale couplings: Initial-boundary-value problem, Nonlinear Anal., 71 (2009), e2487-e2497. doi: 10.1016/j.na.2009.05.047.  Google Scholar

[20]

T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in $\mathbbR^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[21]

T. Kato and G. Ponce, Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces $L_p^s(\mathbbR^2)$, Rev. Mat. Iberoam., 2 (1986), 73-88. doi: 10.4171/RMI/26.  Google Scholar

[22]

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

[23]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Research Notes in Mathematics, Chapman & Hall/CRC, 2002. doi: 10.1201/9781420035674.  Google Scholar

[24]

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

[25]

M. Longaretti, B. Chini, J. W. Jerome and R. Sacco, Electrochemical modeling and characterization of voltage operated channels in nano-bio-electronics, Sensor Letters, 6 (2008), 49-56. doi: 10.1166/sl.2008.010.  Google Scholar

[26]

M. Longaretti, B. Chini, J. W. Jerome and R. Sacco, Computational modeling and simulation of complex systems in bio-electronics, J. Computational Electronics, 7 (2008), 10-13. Google Scholar

[27]

M. Longaretti, G. Marino, B. Chini, J. W. Jerome and R. Sacco, Computational models in nano-bio-electronics: Simulation of ionic transport in voltage operated channels, J. Nanoscience and Nanotechnology, 8 (2008), 3686-3694. Google Scholar

[28]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Commun. Math. Phys., 307 (2011), 713-759. doi: 10.1007/s00220-011-1350-6.  Google Scholar

[29]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584. doi: 10.1016/j.jfa.2012.01.022.  Google Scholar

[30]

F. Planchon, Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 21-23. doi: 10.1016/S0764-4442(00)88138-0.  Google Scholar

[31]

I. Rubinstein, Electro-Diffusion of Ions, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611970814.  Google Scholar

[32]

R. J. Ryham, Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics,, , ().   Google Scholar

[33]

R. J. Ryham, C. Liu and L. Zikatanov, Mathematical models for the deformation of electrolyte droplets, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 649-661. doi: 10.3934/dcdsb.2007.8.649.  Google Scholar

[34]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1015. doi: 10.1142/S0218202509003693.  Google Scholar

[35]

M. Shahinpoor and K. J. Kim, Ionic polymer-metal composites: III. Modeling and simulation as biomimetic sensors, actuators, transducers, and artificial muscles, Smart Mater. Struct., 13 (2004), 1362-1388. doi: 10.1088/0964-1726/13/6/009.  Google Scholar

[36]

H. Triebel, Theory of Function Spaces, Monogr. Math., 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[37]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Commun. Math. Phys., 287 (2009), 211-224. doi: 10.1007/s00220-008-0631-1.  Google Scholar

[38]

T. Zhang and D. Y. Fang, Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations, J. Math. Pures Appl., 90 (2008), 413-449. doi: 10.1016/j.matpur.2008.06.008.  Google Scholar

[39]

J. Zhao, C. Deng and S. Cui, Global well-posedness of a dissipative system arising in electrohydrodynamics in negative-order Besov spaces, J. Math. Physics, 51 (2010), 093101, 17 pp. doi: 10.1063/1.3484184.  Google Scholar

[40]

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

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