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

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

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 $c_0.$
Citation: Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete and 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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