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April  2016, 36(4): 1847-1868. doi: 10.3934/dcds.2016.36.1847

Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows

Hong Cai 1, and  Zhong Tan 1,

1. 

School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and Scientific Computing, Xiamen University, Fujian, Xiamen, 361005, China, China

Received  March 2015 Revised  May 2015 Published  September 2015

In this paper, the compressible magnetohydrodynamic system with some smallness and symmetry assumptions on the time periodic external force is considered in $\mathbb{R}^3$. Based on the uniform estimates and the topological degree theory, we prove the existence of a time periodic solution in a bounded domain. Then by a limiting process, the result in the whole space $\mathbb{R}^3$ is obtained.
Keywords: topological degree theory., $3D$ Compressible magnetohydrodynamic equations, time periodic solution, uniform estimates.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35B10, 76N1.
Citation: Hong Cai, Zhong Tan. Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1847-1868. doi: 10.3934/dcds.2016.36.1847
References:
[1]

J. Březina and K. Kagei, Decay properties of solutions to the linearized compressible Navier-Stokes equation around time-periodic parallel flow, Math. Models Methods Appl. Sci., 22 (2012), 1250007, 53 pp. doi: 10.1142/S0218202512500078.  Google Scholar

[2]

J. Březina and K. Kagei, Spectral properties of the linearized compressible Navier-Stokes equation around time-periodic parallel flow, J. Differential Equations, 255 (2013), 1132-1195. doi: 10.1016/j.jde.2013.04.036.  Google Scholar

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

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

[5]

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

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J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875. doi: 10.1016/j.jde.2014.01.021.  Google Scholar

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J. Fan and W. 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. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. Google Scholar

[9]

J. Fan and K. Zhao, Global Cauchy problem of $2D$ generalized magnetohydrodynamic equations, J. Math. Anal. Appl., 420 (2014), 1024-1032. doi: 10.1016/j.jmaa.2014.06.030.  Google Scholar

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E. Feireisl, P. B. Mucha, A. Novotny and M. Pokorny, Time-periodic solutions to the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 745-786. doi: 10.1007/s00205-012-0492-9.  Google Scholar

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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

[12]

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

[13]

C. H. Jin and T. Yang, Periodic solutions for a $3-D$ compressible Navier-Stokes equations in a periodic domain,, submitted to JDE., ().   Google Scholar

[14]

C. H. Jin and T. Yang, Time periodic solutions to $3-D$ compressible Navier-Stokes system with external force,, submitted., ().   Google Scholar

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Y. Kagei and K. Tsuda, Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry, J. Differential Equations, 258 (2015), 399-444. doi: 10.1016/j.jde.2014.09.016.  Google Scholar

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

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

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H. L. Li, X. Y. Xu and J. W. Zhang, Global Classical Solutions to $3D$ Compressible Magnetohydrodynamic Equations with Large Oscillations and Vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387. doi: 10.1137/120893355.  Google Scholar

[19]

H. F. Ma, S. Ukai and T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), 2275-2293. doi: 10.1016/j.jde.2009.11.031.  Google Scholar

[20]

A. Matsumura and T. Nishida, Periodic solutions of a viscous gas equation, Recent topics in nonlinear PDE, IV (Kyoto, 1988), 160 (1982), 49-82. doi: 10.1016/S0304-0208(08)70506-1.  Google Scholar

[21]

E. A. Notte, M. D. Rojas and M. A. Rojas, Periodic strong solutions of the magnetohydrodynamic type equations, Proyecciones, 21 (2002), 199-224. doi: 10.4067/S0716-09172002000300001.  Google Scholar

[22]

Z. Tan and H. Q. Wang, Time periodic solutions of compressible magnetohydrodynamic equations, Nonlinear Anal., 76 (2013), 153-164. doi: 10.1016/j.na.2012.08.012.  Google Scholar

[23]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647.  Google Scholar

[24]

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

[25]

W. Yan and Y. Li, Existence of periodic flows for compressible Magnetohydrodynamics in $\mathbbT^3$,, Submitted., ().   Google Scholar

[26]

Y. F. Yang, X. H. Gu and C. S. Dou, Global well-posedness of strong solutions to the magnetohydrodynamic equations of compressible flows, Nonlinear Anal., 95 (2014), 23-37. doi: 10.1016/j.na.2013.08.024.  Google Scholar

show all references

References:
[1]

J. Březina and K. Kagei, Decay properties of solutions to the linearized compressible Navier-Stokes equation around time-periodic parallel flow, Math. Models Methods Appl. Sci., 22 (2012), 1250007, 53 pp. doi: 10.1142/S0218202512500078.  Google Scholar

[2]

J. Březina and K. Kagei, Spectral properties of the linearized compressible Navier-Stokes equation around time-periodic parallel flow, J. Differential Equations, 255 (2013), 1132-1195. doi: 10.1016/j.jde.2013.04.036.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

J. Fan, F. Li, G. Nakamura and Z. Tan, Regularity criteria for the three-dimensional magnetohydrodynamic equations, J. Differential Equations, 256 (2014), 2858-2875. doi: 10.1016/j.jde.2014.01.021.  Google Scholar

[7]

J. Fan and W. 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

[8]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. Google Scholar

[9]

J. Fan and K. Zhao, Global Cauchy problem of $2D$ generalized magnetohydrodynamic equations, J. Math. Anal. Appl., 420 (2014), 1024-1032. doi: 10.1016/j.jmaa.2014.06.030.  Google Scholar

[10]

E. Feireisl, P. B. Mucha, A. Novotny and M. Pokorny, Time-periodic solutions to the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 745-786. doi: 10.1007/s00205-012-0492-9.  Google Scholar

[11]

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

[12]

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

[13]

C. H. Jin and T. Yang, Periodic solutions for a $3-D$ compressible Navier-Stokes equations in a periodic domain,, submitted to JDE., ().   Google Scholar

[14]

C. H. Jin and T. Yang, Time periodic solutions to $3-D$ compressible Navier-Stokes system with external force,, submitted., ().   Google Scholar

[15]

Y. Kagei and K. Tsuda, Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry, J. Differential Equations, 258 (2015), 399-444. doi: 10.1016/j.jde.2014.09.016.  Google Scholar

[16]

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

[17]

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

[18]

H. L. Li, X. Y. Xu and J. W. Zhang, Global Classical Solutions to $3D$ Compressible Magnetohydrodynamic Equations with Large Oscillations and Vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387. doi: 10.1137/120893355.  Google Scholar

[19]

H. F. Ma, S. Ukai and T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), 2275-2293. doi: 10.1016/j.jde.2009.11.031.  Google Scholar

[20]

A. Matsumura and T. Nishida, Periodic solutions of a viscous gas equation, Recent topics in nonlinear PDE, IV (Kyoto, 1988), 160 (1982), 49-82. doi: 10.1016/S0304-0208(08)70506-1.  Google Scholar

[21]

E. A. Notte, M. D. Rojas and M. A. Rojas, Periodic strong solutions of the magnetohydrodynamic type equations, Proyecciones, 21 (2002), 199-224. doi: 10.4067/S0716-09172002000300001.  Google Scholar

[22]

Z. Tan and H. Q. Wang, Time periodic solutions of compressible magnetohydrodynamic equations, Nonlinear Anal., 76 (2013), 153-164. doi: 10.1016/j.na.2012.08.012.  Google Scholar

[23]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647.  Google Scholar

[24]

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

[25]

W. Yan and Y. Li, Existence of periodic flows for compressible Magnetohydrodynamics in $\mathbbT^3$,, Submitted., ().   Google Scholar

[26]

Y. F. Yang, X. H. Gu and C. S. Dou, Global well-posedness of strong solutions to the magnetohydrodynamic equations of compressible flows, Nonlinear Anal., 95 (2014), 23-37. doi: 10.1016/j.na.2013.08.024.  Google Scholar

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