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

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].
Citation: Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete and 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[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.

[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.

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