American Institute of Mathematical Sciences

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February  2019, 12(1): 37-58. doi: 10.3934/krm.2019002

Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density

Fei Chen 1, , Boling Guo 2, and  Xiaoping Zhai 3,,

1. 

School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, China
2. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
3. 

School of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong 518060, China

* Corresponding author: Xiaoping Zhai

Received  March 2017 Revised  February 2018 Published  July 2018

Fund Project: The first author is supported by the Postdoctoral Science Foundation of China grant 2017M620688, the second author is supported by NSFC grant 11731014, 11571254 and the third author is supported by NSFC grant 11601533.

In this paper, we consider the Cauchy problem of the incompressible MHD system with discontinuous initial density in ${\mathbb R}^3$. We establish the global well-posedness of the MHD system if the initial data satisfies
$(ρ_0, u_0, H_0)∈ L^{∞}({\mathbb R}^3)× H^s({\mathbb R}^3)× H^s({\mathbb R}^3)$
 with
$\frac{1}{2} < s \le 1$
 and
$0 < \underline{ρ} \le ρ_0 \le \overline{ρ} < +∞,~~~~ \|(u_0, H_0)\|_{\dot{H}^{\frac 12}} \le c, $
for some small
$c>0$
 which only depends on
$\underline{ρ}, \overline{ρ}$
. As a byproduct, we also get the decay estimate of the solution.
Keywords: Global well-posedness, incompressible MHD system, Cauchy problem, decay estimate.
Mathematics Subject Classification: Primary: 35Q35, 76D03; Secondary: 35B40.
Citation: Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002
References:
[1]

H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3-D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 832-881.  doi: 10.1002/cpa.20351.  Google Scholar

[2]

H. Abidi, G. Gui and P. Zhang, On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Rational Mech. Anal., 204 (2012), 189-230.  doi: 10.1007/s00205-011-0473-4.  Google Scholar

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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

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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

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[9]

D. Chen, Z. Zhang and W. Zhao, Fujita-Kato theorem for the 3-D inhomogenous Navier-Stokes equations, J. Differential Equations, 261 (2016), 738-761.  doi: 10.1016/j.jde.2016.03.024.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

R. Danchin and P. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409.  Google Scholar

[14]

R. Danchin and P. B. Mucha, Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207 (2013), 991-1023.  doi: 10.1007/s00205-012-0586-4.  Google Scholar

[15]

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

[16]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.  Google Scholar

[17]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, AMS, Providence, RI, 1998.  Google Scholar

[18]

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

[19]

H. Gong and J. Li, Global existence of strong solutions to incompressible MHD, Commun. Pure Appl. Anal., 13 (2014), 1337-1345.  doi: 10.3934/cpaa.2014.13.1337.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

D. Hoff, Global solutions of the Navier-Stokes equations for mutidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.  Google Scholar

[24]

D. Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions, Comm. Pure Appl. Math., 55 (2002), 1365-1407.  doi: 10.1002/cpa.10046.  Google Scholar

[25]

X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimentional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar

[26]

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

[27]

J. Jia, J. Peng and K. Li, On the decay and stability of global solutions to the 3D inhomogenous MHD system, Comm. Pure Appl. Anal., 16 (2017), 745-780.  doi: 10.3934/cpaa.2017036.  Google Scholar

[28]

A. V. Kazhikhov, Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, Dokl. Akad. Nauk SSSR, 216 (1974), 1008-1010.   Google Scholar

[29]

F. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034.  Google Scholar

[30]

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

[31]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3, Oxford Sci. Publ., The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[32]

M. Paicu, P. Zhang and Z. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with boundary density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.  doi: 10.1080/03605302.2013.780079.  Google Scholar

[33]

X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magentic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020.  Google Scholar

[34]

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

[35]

H. Xu, Y. Li and X. Zhai, On the well-posedness of 2-D incompressible Navier-Stokes equations with variable viscosity in critical spaces, J. Differential Equations, 260 (2016), 6604-6637.  doi: 10.1016/j.jde.2016.01.007.  Google Scholar

[36]

X. Zhai, Y. Li and H. Xu, Global well-posedness for the 2-D nonhomogeneous incompressible MHD equations with large initial data, Nonlinear Anal. Real World Appl., 33 (2017), 1-18.  doi: 10.1016/j.nonrwa.2016.05.009.  Google Scholar

[37]

X. Zhai, Y. Li and W. Yan, Global well-posedness for the 3-D incompressible inhomogeneous MHD system in the ciritical Besov spaces, J. Math. Anal. Appl., 432 (2015), 179-195.  doi: 10.1016/j.jmaa.2015.06.048.  Google Scholar

[38]

X. Zhai and Z. Yin, Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations, J. Differential Equations, 262 (2017), 1359-1412.  doi: 10.1016/j.jde.2016.10.016.  Google Scholar

show all references

References:
[1]

H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3-D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 832-881.  doi: 10.1002/cpa.20351.  Google Scholar

[2]

H. Abidi, G. Gui and P. Zhang, On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Rational Mech. Anal., 204 (2012), 189-230.  doi: 10.1007/s00205-011-0473-4.  Google Scholar

[3]

H. Abidi and T. Hmidi, Résultats d'existence dans des espaces critiques pour le systéme de la MHD inhomogéne, Ann. Math. Blaise Pascal, 14 (2007), 103-148.   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

F. Chen, Y. Li and H. Xu, Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data, Discrete Contin. Dyn. Syst., 36 (2016), 2945-2967.  doi: 10.3934/dcds.2016.36.2945.  Google Scholar

[9]

D. Chen, Z. Zhang and W. Zhao, Fujita-Kato theorem for the 3-D inhomogenous Navier-Stokes equations, J. Differential Equations, 261 (2016), 738-761.  doi: 10.1016/j.jde.2016.03.024.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

R. Danchin and P. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409.  Google Scholar

[14]

R. Danchin and P. B. Mucha, Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207 (2013), 991-1023.  doi: 10.1007/s00205-012-0586-4.  Google Scholar

[15]

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

[16]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.  Google Scholar

[17]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, AMS, Providence, RI, 1998.  Google Scholar

[18]

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

[19]

H. Gong and J. Li, Global existence of strong solutions to incompressible MHD, Commun. Pure Appl. Anal., 13 (2014), 1337-1345.  doi: 10.3934/cpaa.2014.13.1337.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

D. Hoff, Global solutions of the Navier-Stokes equations for mutidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.  Google Scholar

[24]

D. Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions, Comm. Pure Appl. Math., 55 (2002), 1365-1407.  doi: 10.1002/cpa.10046.  Google Scholar

[25]

X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimentional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar

[26]

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

[27]

J. Jia, J. Peng and K. Li, On the decay and stability of global solutions to the 3D inhomogenous MHD system, Comm. Pure Appl. Anal., 16 (2017), 745-780.  doi: 10.3934/cpaa.2017036.  Google Scholar

[28]

A. V. Kazhikhov, Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, Dokl. Akad. Nauk SSSR, 216 (1974), 1008-1010.   Google Scholar

[29]

F. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034.  Google Scholar

[30]

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

[31]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3, Oxford Sci. Publ., The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[32]

M. Paicu, P. Zhang and Z. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with boundary density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.  doi: 10.1080/03605302.2013.780079.  Google Scholar

[33]

X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magentic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020.  Google Scholar

[34]

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

[35]

H. Xu, Y. Li and X. Zhai, On the well-posedness of 2-D incompressible Navier-Stokes equations with variable viscosity in critical spaces, J. Differential Equations, 260 (2016), 6604-6637.  doi: 10.1016/j.jde.2016.01.007.  Google Scholar

[36]

X. Zhai, Y. Li and H. Xu, Global well-posedness for the 2-D nonhomogeneous incompressible MHD equations with large initial data, Nonlinear Anal. Real World Appl., 33 (2017), 1-18.  doi: 10.1016/j.nonrwa.2016.05.009.  Google Scholar

[37]

X. Zhai, Y. Li and W. Yan, Global well-posedness for the 3-D incompressible inhomogeneous MHD system in the ciritical Besov spaces, J. Math. Anal. Appl., 432 (2015), 179-195.  doi: 10.1016/j.jmaa.2015.06.048.  Google Scholar

[38]

X. Zhai and Z. Yin, Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations, J. Differential Equations, 262 (2017), 1359-1412.  doi: 10.1016/j.jde.2016.10.016.  Google Scholar

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