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

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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

Abstract
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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 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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