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Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density
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 |
$(ρ_0, u_0, H_0)∈ L^{∞}({\mathbb R}^3)× H^s({\mathbb R}^3)× H^s({\mathbb R}^3)$ |
$\frac{1}{2} < s \le 1$ |
$0 < \underline{ρ} \le ρ_0 \le \overline{ρ} < +∞,~~~~ \|(u_0, H_0)\|_{\dot{H}^{\frac 12}} \le c, $ |
$c>0$ |
$\underline{ρ}, \overline{ρ}$ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
B. Desjardins and C. Le Bris,
Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394.
|
[16] |
G. Duvaut and J. L. Lions,
Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[17] |
L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, AMS, Providence, RI, 1998. |
[18] |
J. F. Gerbeau and C. Le Bris,
Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
B. Desjardins and C. Le Bris,
Remarks on a nonhomogeneous model of magnetohydrodynamics, Differential Integral Equations, 11 (1998), 377-394.
|
[16] |
G. Duvaut and J. L. Lions,
Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.
doi: 10.1007/BF00250512. |
[17] |
L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, AMS, Providence, RI, 1998. |
[18] |
J. F. Gerbeau and C. Le Bris,
Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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