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

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  • 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.
    Mathematics Subject Classification: Primary: 35K15, 35K55; Secondary: 35Q35, 76A05.


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  • [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.


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


    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.


    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.


    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.


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


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.doi: 10.1006/aima.2000.1937.


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


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


    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.


    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.


    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.


    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.


    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.


    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.


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


    R. J. Ryham, Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics, arXiv:0910.4973v1.


    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.


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


    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.


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


    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.


    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.


    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.


    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.

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