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October  2016, 36(10): 5817-5835. doi: 10.3934/dcds.2016056

On the global well-posedness to the 3-D Navier-Stokes-Maxwell system

Gaocheng Yue 1, and  Chengkui Zhong 2,

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106
2. 

Department of Mathematics, Nanjing University, Nanjing 210093

Received  September 2015 Revised  March 2016 Published  July 2016

The present paper is devoted to the well-posedness issue of solutions of a full system of the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations. By means of Littlewood-Paley analysis we prove the global well-posedness of solutions in the Besov spaces $\dot{B}_{2,1}^\frac1{2}\times B_{2,1}^\frac3{2}\times B_{2,1}^\frac3{2}$ provided the norm of initial data is small enough in the sense that \begin{align*} \big(\|u_0^h\|_{\dot{B}_{2,1}^\frac1{2}} +\|E_0\|_{B_{2,1}^\frac{3}{2}}+\|B_0\|_{B_{2,1}^\frac{3}{2}}\big)\exp \Big\{\frac{C_0}{\nu^2}\|u_0^3\|_{\dot{B}_{2,1}^\frac1{2}}^2\Big\}\leq c_0, \end{align*} for some sufficiently small constant $c_0.$
Keywords: Navier-Stokes-Maxwell system, Besov spaces., Littlewood-Paley theory, Well-posedness.
Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 76A0.
Citation: Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056
References:
[1]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

J. Chemin, Localization in Fourier space and Navier-Stokes system Phase space analysis of Partial Differential Equations, Pubbl. Cert. Ric. Mat. Ennio de Gorg Scuola Norma. Sup. Pisa, \textbfI (2004), 53-135.  Google Scholar

[3]

J. 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

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J. 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

[5]

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

[6]

P. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333.  Google Scholar

[7]

P. Germain, S. Ibrahim and N. Masmoudi, Well-posedness of the Navier-Stokes-Maxwell equations, Proceedings of the Royal Society of Edinburgh, 144 (2014), 71-86. doi: 10.1017/S0308210512001242.  Google Scholar

[8]

G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284. doi: 10.1016/j.aim.2010.03.022.  Google Scholar

[9]

S. Ibrahim and S. Keraani, Global small solutions for the coupled Navier-Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295. doi: 10.1137/100819813.  Google Scholar

[10]

S. Ibrahim and T. Yoneda, Local solvability and loss of smoothness of the Navier-Stokes- Maxwell equations with large initial data, J. Math. Analysis Applic., 396 (2012), 555-561. doi: 10.1016/j.jmaa.2012.06.038.  Google Scholar

[11]

N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571. doi: 10.1016/j.matpur.2009.08.007.  Google Scholar

[12]

M. Paicu and P. Zhang, Global solutions to the 3-D 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

[13]

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

[14]

T. Zhang, Global wellposed 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

show all references

References:
[1]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

J. Chemin, Localization in Fourier space and Navier-Stokes system Phase space analysis of Partial Differential Equations, Pubbl. Cert. Ric. Mat. Ennio de Gorg Scuola Norma. Sup. Pisa, \textbfI (2004), 53-135.  Google Scholar

[3]

J. 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

[4]

J. 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

[5]

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

[6]

P. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511626333.  Google Scholar

[7]

P. Germain, S. Ibrahim and N. Masmoudi, Well-posedness of the Navier-Stokes-Maxwell equations, Proceedings of the Royal Society of Edinburgh, 144 (2014), 71-86. doi: 10.1017/S0308210512001242.  Google Scholar

[8]

G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284. doi: 10.1016/j.aim.2010.03.022.  Google Scholar

[9]

S. Ibrahim and S. Keraani, Global small solutions for the coupled Navier-Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295. doi: 10.1137/100819813.  Google Scholar

[10]

S. Ibrahim and T. Yoneda, Local solvability and loss of smoothness of the Navier-Stokes- Maxwell equations with large initial data, J. Math. Analysis Applic., 396 (2012), 555-561. doi: 10.1016/j.jmaa.2012.06.038.  Google Scholar

[11]

N. Masmoudi, Global well posedness for the Maxwell-Navier-Stokes system in 2D, J. Math. Pures Appl., 93 (2010), 559-571. doi: 10.1016/j.matpur.2009.08.007.  Google Scholar

[12]

M. Paicu and P. Zhang, Global solutions to the 3-D 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

[13]

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

[14]

T. Zhang, Global wellposed 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

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