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June  2017, 37(6): 3435-3465. doi: 10.3934/dcds.2017146

Asymptotic stability of stationary solutions for magnetohydrodynamic equations

Zhong Tan and  Leilei Tong

School of Mathematical Science and Fujian Provincial Key Laboratory, on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen 361005, China

Received  July 2015 Revised  January 2017 Published  February 2017

Fund Project: Supported by the National Natural Science Foundation of China (Grant No. 11271305,11531010) and the Fundamental Research Funds for Xiamen University (No. 201412G004).

In this paper, we are concerned with the compressible magnetohydrodynamic equations with Coulomb force in three-dimensional space. We show the asymptotic stability of solutions to the Cauchy problem near the non-constant equilibrium state provided that the initial perturbation is sufficiently small. Moreover, the convergence rates are obtained by combining the linear Lp-Lq decay estimates and the higher-order energy estimates.

Keywords: MHD equations, global solutions, stability, energy method, time decay rate.
Mathematics Subject Classification: Primary: 35B35, 35M31; Secondary: 35Q60, 35Q35.
Citation: Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146
References:
[1]

M. Alessandro, T. Yuri and T. Paola, Well-posedness of the linearized problem for MHD contact discontinuities, J. Differential Equations, 258 (2015), 2531-2571.  doi: 10.1016/j.jde.2014.12.018.  Google Scholar

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Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydroynamics equations, Nonlinear Anal., 72 (2010), 4438-4451.  doi: 10.1016/j.na.2010.02.019.  Google Scholar

[3]

G. Q. Chen and D. H. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

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G. Q. Chen and D. H. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.  doi: 10.1007/s00033-003-1017-z.  Google Scholar

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L. L. Du and D. Q. Zhou, Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal., 47 (2015), 1562-1589.  doi: 10.1137/140959821.  Google Scholar

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J.S. Fan, A. Ahmed, H. Tasawar, N. Gen and Y. Zhou, On strong solutions to the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 22 (2015), 423-434.  doi: 10.1016/j.nonrwa.2014.10.003.  Google Scholar

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J. S. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1.  Google Scholar

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J. S. Fan and W. H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.  doi: 10.1016/j.na.2007.10.005.  Google Scholar

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J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.  Google Scholar

[10]

E. Feireisl, A. Novotný and H. Petleltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

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Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.  Google Scholar

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[13]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.  doi: 10.1007/s00033-005-4057-8.  Google Scholar

[14]

X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.  Google Scholar

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X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294.  doi: 10.1137/080723983.  Google Scholar

[16]

X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.  Google Scholar

[17]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary condtions, Comm. Math. Phys., 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.  Google Scholar

[18]

S. Jiang, Q. C. Ju, F. C. Li and Z. P. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.  Google Scholar

[19]

Q. C. Ju, F. C. Li and Y. Li, Asymptotic limits of the full comressible magnetohydrodynamic equations, SIAM J. Math. Anal., 45 (2013), 2597-2624.  doi: 10.1137/130913390.  Google Scholar

[20]

N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys., 251 (2004), 365-376.  doi: 10.1007/s00220-004-1062-2.  Google Scholar

[21]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagneto fluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.  Google Scholar

[22]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad., 58 (1982), 384-387.  doi: 10.3792/pjaa.58.384.  Google Scholar

[23]

T. Kobayashi and T. Suzuki, Weak solutions to the Navier-Stokes-Poisson equation, Adv. Math. Sci. Appl., 18 (2008), 141-168.   Google Scholar

[24]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar

[25]

H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in ℝ3, Arch. Ration. Mech. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4.  Google Scholar

[26]

H. L. Li, X. Y. Xu and J. W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillationa and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.  Google Scholar

[27]

F. C. Li and H. J. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect., 141 (2011), 109-126.  doi: 10.1017/S0308210509001632.  Google Scholar

[28] P. L. Lions, Mathematical Topics in Fluids Mechanics. Oxford Lecture Ser. Math. Appl, Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[29] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.   Google Scholar
[30]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[31]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.  Google Scholar

[32]

Z. Tan, L. L. Tong and Y. Wang, Large time behavior of the compressible magnetohydrodynamic equations with Coulomb force, J. Math. Anal. Appl., 427 (2015), 600-617.  doi: 10.1016/j.jmaa.2015.02.077.  Google Scholar

[33]

Z. Tan and Y. J. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.  doi: 10.1016/j.na.2009.05.012.  Google Scholar

[34]

Z. Tan and G. C. Wu, Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions, Nonlinear Anal. Real World Appl., 13 (2012), 650-664.  doi: 10.1016/j.nonrwa.2011.08.005.  Google Scholar

[35]

Z. Tan and H. Q. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl., 14 (2013), 188-201.  doi: 10.1016/j.nonrwa.2012.05.012.  Google Scholar

[36]

Z. Tan, Y. J. Wang and Y. Wang, Stability of steady states of the Navier-Stokes-Poisson equations with non-flat doping profile, SIAM J. Math. Anal., 47 (2015), 179-209.  doi: 10.1137/130950069.  Google Scholar

[37]

Z. Tan, Y. Wang and X. Zhang, Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, Kinet. Relat. Models, 5 (2012), 615-638.  doi: 10.3934/krm.2012.5.615.  Google Scholar

[38]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.  Google Scholar

[39]

D. H. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.  doi: 10.1137/S0036139902409284.  Google Scholar

[40]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297.  doi: 10.1016/j.jde.2012.03.006.  Google Scholar

[41]

L. Zhang and S. Li, A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion, Nonlinear Anal. Real World Appl., 21 (2015), 197-206.  doi: 10.1016/j.nonrwa.2014.07.005.  Google Scholar

[42]

G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, J. Differential Equations, 250 (2011), 866-891.  doi: 10.1016/j.jde.2010.07.035.  Google Scholar

show all references

References:
[1]

M. Alessandro, T. Yuri and T. Paola, Well-posedness of the linearized problem for MHD contact discontinuities, J. Differential Equations, 258 (2015), 2531-2571.  doi: 10.1016/j.jde.2014.12.018.  Google Scholar

[2]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydroynamics equations, Nonlinear Anal., 72 (2010), 4438-4451.  doi: 10.1016/j.na.2010.02.019.  Google Scholar

[3]

G. Q. Chen and D. H. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

[4]

G. Q. Chen and D. H. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys., 54 (2003), 608-632.  doi: 10.1007/s00033-003-1017-z.  Google Scholar

[5]

L. L. Du and D. Q. Zhou, Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal., 47 (2015), 1562-1589.  doi: 10.1137/140959821.  Google Scholar

[6]

J.S. Fan, A. Ahmed, H. Tasawar, N. Gen and Y. Zhou, On strong solutions to the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 22 (2015), 423-434.  doi: 10.1016/j.nonrwa.2014.10.003.  Google Scholar

[7]

J. S. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys., 270 (2007), 691-708.  doi: 10.1007/s00220-006-0167-1.  Google Scholar

[8]

J. S. Fan and W. H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660.  doi: 10.1016/j.na.2007.10.005.  Google Scholar

[9]

J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.  Google Scholar

[10]

E. Feireisl, A. Novotný and H. Petleltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[11]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.  Google Scholar

[12]

C. C. Hao and H. L. Li, Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equations, 246 (2009), 4791-4812.  doi: 10.1016/j.jde.2008.11.019.  Google Scholar

[13]

D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 791-804.  doi: 10.1007/s00033-005-4057-8.  Google Scholar

[14]

X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284.  doi: 10.1007/s00220-008-0497-2.  Google Scholar

[15]

X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294.  doi: 10.1137/080723983.  Google Scholar

[16]

X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.  Google Scholar

[17]

S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary condtions, Comm. Math. Phys., 297 (2010), 371-400.  doi: 10.1007/s00220-010-0992-0.  Google Scholar

[18]

S. Jiang, Q. C. Ju, F. C. Li and Z. P. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.  Google Scholar

[19]

Q. C. Ju, F. C. Li and Y. Li, Asymptotic limits of the full comressible magnetohydrodynamic equations, SIAM J. Math. Anal., 45 (2013), 2597-2624.  doi: 10.1137/130913390.  Google Scholar

[20]

N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys., 251 (2004), 365-376.  doi: 10.1007/s00220-004-1062-2.  Google Scholar

[21]

S. Kawashima, Smooth global solutions for two-dimensional equations of electromagneto fluid dynamics, Japan J. Appl. Math., 1 (1984), 207-222.  doi: 10.1007/BF03167869.  Google Scholar

[22]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad., 58 (1982), 384-387.  doi: 10.3792/pjaa.58.384.  Google Scholar

[23]

T. Kobayashi and T. Suzuki, Weak solutions to the Navier-Stokes-Poisson equation, Adv. Math. Sci. Appl., 18 (2008), 141-168.   Google Scholar

[24]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.  Google Scholar

[25]

H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in ℝ3, Arch. Ration. Mech. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4.  Google Scholar

[26]

H. L. Li, X. Y. Xu and J. W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillationa and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.  Google Scholar

[27]

F. C. Li and H. J. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect., 141 (2011), 109-126.  doi: 10.1017/S0308210509001632.  Google Scholar

[28] P. L. Lions, Mathematical Topics in Fluids Mechanics. Oxford Lecture Ser. Math. Appl, Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[29] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.   Google Scholar
[30]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[31]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.  Google Scholar

[32]

Z. Tan, L. L. Tong and Y. Wang, Large time behavior of the compressible magnetohydrodynamic equations with Coulomb force, J. Math. Anal. Appl., 427 (2015), 600-617.  doi: 10.1016/j.jmaa.2015.02.077.  Google Scholar

[33]

Z. Tan and Y. J. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.  doi: 10.1016/j.na.2009.05.012.  Google Scholar

[34]

Z. Tan and G. C. Wu, Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions, Nonlinear Anal. Real World Appl., 13 (2012), 650-664.  doi: 10.1016/j.nonrwa.2011.08.005.  Google Scholar

[35]

Z. Tan and H. Q. Wang, Optimal decay rates of the compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl., 14 (2013), 188-201.  doi: 10.1016/j.nonrwa.2012.05.012.  Google Scholar

[36]

Z. Tan, Y. J. Wang and Y. Wang, Stability of steady states of the Navier-Stokes-Poisson equations with non-flat doping profile, SIAM J. Math. Anal., 47 (2015), 179-209.  doi: 10.1137/130950069.  Google Scholar

[37]

Z. Tan, Y. Wang and X. Zhang, Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, Kinet. Relat. Models, 5 (2012), 615-638.  doi: 10.3934/krm.2012.5.615.  Google Scholar

[38]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.  Google Scholar

[39]

D. H. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math., 63 (2003), 1424-1441.  doi: 10.1137/S0036139902409284.  Google Scholar

[40]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297.  doi: 10.1016/j.jde.2012.03.006.  Google Scholar

[41]

L. Zhang and S. Li, A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion, Nonlinear Anal. Real World Appl., 21 (2015), 197-206.  doi: 10.1016/j.nonrwa.2014.07.005.  Google Scholar

[42]

G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in ℝ3, J. Differential Equations, 250 (2011), 866-891.  doi: 10.1016/j.jde.2010.07.035.  Google Scholar

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