Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35Q35; Secondary: 35B10, 76N10.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


    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.


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


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


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


    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.


    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.


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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(193) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint