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Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows
1. | School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and Scientific Computing, Xiamen University, Fujian, Xiamen, 361005, China, China |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. |
[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. |
[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. |
[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. |
[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. |
[13] |
C. H. Jin and T. Yang, Periodic solutions for a $3-D$ compressible Navier-Stokes equations in a periodic domain, submitted to JDE. |
[14] |
C. H. Jin and T. Yang, Time periodic solutions to $3-D$ compressible Navier-Stokes system with external force, submitted. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
W. Yan and Y. Li, Existence of periodic flows for compressible Magnetohydrodynamics in $\mathbbT^3$, Submitted. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. |
[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. |
[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. |
[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. |
[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. |
[13] |
C. H. Jin and T. Yang, Periodic solutions for a $3-D$ compressible Navier-Stokes equations in a periodic domain, submitted to JDE. |
[14] |
C. H. Jin and T. Yang, Time periodic solutions to $3-D$ compressible Navier-Stokes system with external force, submitted. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
W. Yan and Y. Li, Existence of periodic flows for compressible Magnetohydrodynamics in $\mathbbT^3$, Submitted. |
[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. |
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