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June  2016, 36(6): 2945-2967. doi: 10.3934/dcds.2016.36.2945

Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data

Fei Chen 1, , Yongsheng Li 1, and  Huan Xu 1,

1. 

Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China, China

Received  April 2015 Revised  September 2015 Published  December 2015

In this paper, we show the 3D nonhomogeneous incompressible MHD equations have a global solution provided that the initial data in critical Besov spaces $\dot{B}_{q,1}^{{3}/{q}}(\mathbb{R}^{3})\times\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3}) \times\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3})$ satisfy a nonlinear smallness condition for all $1< q\leq p<6$, ${1}/{q}-{1}/{p}<{1}/{3}$ if the initial density is near a positive constant. Moreover, this solution is unique under the restriction condition ${1}/{p}+{1}/{q}\geq{2}/{3}$. Motivated by Chemin and Gallagher [7], we also provide an example of initial data satisfying that nonlinear smallness condition, but the norms of $u_{0},b_{0}$ (even all their components) can be arbitrarily large in $\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3})$. In particular, when $b$ identically equals 0, our results improve that of Paicu and Zhang [28].
Keywords: critical Besov spaces., Littlewood-Paley theory, Cauchy problem, global well-posedness, Nonhomogeneous MHD equations.
Mathematics Subject Classification: Primary: 35Q35, 76D03; Secondary: 35B4.
Citation: Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945
References:
[1]

H. Abidi and T. Hmidi, Résultats d'existence dans des espaces critiques pour le système de la MHD inhomogène, (French) [Existence in critical spaces for the inhomogeneous MHD system], Ann. Math. Blaise Pascal, 14 (2007), 103-148. doi: 10.5802/ambp.230.  Google Scholar

[2]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476. doi: 10.1017/S0308210506001181.  Google Scholar

[3]

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

[4]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[5]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[6]

J. Y. Chemin, Perfect Incompressible Fluids, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[7]

J. Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in $\mathbfR^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624. doi: 10.1016/j.anihpc.2007.05.008.  Google Scholar

[8]

Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 275 (2007), 861-872. doi: 10.1007/s00220-007-0319-y.  Google Scholar

[9]

Q. Chen, Z. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107. doi: 10.1002/mma.1338.  Google Scholar

[10]

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

[11]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.  Google Scholar

[12]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[13]

B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394.  Google Scholar

[14]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[15]

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

[16]

J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452.  Google Scholar

[17]

J. F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for The Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.  Google Scholar

[18]

G. Gui, Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity, J. Funct. Anal., 267 (2014), 1488-1539. doi: 10.1016/j.jfa.2014.06.002.  Google Scholar

[19]

C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 238 (2007), 1-17. doi: 10.1016/j.jde.2007.03.023.  Google Scholar

[20]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.  Google Scholar

[21]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.  Google Scholar

[22]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[23]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, $2^{nd}$ edition, Butterworth-Heinemann, U.K., 1999. Google Scholar

[24]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.  Google Scholar

[25]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[26]

R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers Group, Dordrecht, 1990. doi: 10.1007/978-94-015-7883-7.  Google Scholar

[27]

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

[28]

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

[29]

J. Peetre, New Thoughts on Besov Spaces, Duke University, Durham, N.C., 1976.  Google Scholar

[30]

R. V. Polovin and V. P. Demutskiĭ, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990. Google Scholar

[31]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar

[32]

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 (2015), 091512, 18 pp. doi: 10.1063/1.4931467.  Google Scholar

[33]

Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1.  Google Scholar

show all references

References:
[1]

H. Abidi and T. Hmidi, Résultats d'existence dans des espaces critiques pour le système de la MHD inhomogène, (French) [Existence in critical spaces for the inhomogeneous MHD system], Ann. Math. Blaise Pascal, 14 (2007), 103-148. doi: 10.5802/ambp.230.  Google Scholar

[2]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476. doi: 10.1017/S0308210506001181.  Google Scholar

[3]

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

[4]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[5]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[6]

J. Y. Chemin, Perfect Incompressible Fluids, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[7]

J. Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in $\mathbfR^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624. doi: 10.1016/j.anihpc.2007.05.008.  Google Scholar

[8]

Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 275 (2007), 861-872. doi: 10.1007/s00220-007-0319-y.  Google Scholar

[9]

Q. Chen, Z. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107. doi: 10.1002/mma.1338.  Google Scholar

[10]

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

[11]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.  Google Scholar

[12]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[13]

B. Desjardins and C. Le Bris, Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394.  Google Scholar

[14]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[15]

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

[16]

J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452.  Google Scholar

[17]

J. F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for The Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.  Google Scholar

[18]

G. Gui, Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity, J. Funct. Anal., 267 (2014), 1488-1539. doi: 10.1016/j.jfa.2014.06.002.  Google Scholar

[19]

C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 238 (2007), 1-17. doi: 10.1016/j.jde.2007.03.023.  Google Scholar

[20]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.  Google Scholar

[21]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.  Google Scholar

[22]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[23]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, $2^{nd}$ edition, Butterworth-Heinemann, U.K., 1999. Google Scholar

[24]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.  Google Scholar

[25]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[26]

R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers Group, Dordrecht, 1990. doi: 10.1007/978-94-015-7883-7.  Google Scholar

[27]

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

[28]

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

[29]

J. Peetre, New Thoughts on Besov Spaces, Duke University, Durham, N.C., 1976.  Google Scholar

[30]

R. V. Polovin and V. P. Demutskiĭ, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990. Google Scholar

[31]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar

[32]

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 (2015), 091512, 18 pp. doi: 10.1063/1.4931467.  Google Scholar

[33]

Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1.  Google Scholar

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