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

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.
Citation: Hong Cai, Zhong Tan. Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - A, 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). 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. 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. 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. 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. 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. 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. 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. Google Scholar

[9]

J. Fan and K. Zhao, Global Cauchy problem of $2D$ generalized magnetohydrodynamic equations,, J. Math. Anal. Appl., 420 (2014), 1024. 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. 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. 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. 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. 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. 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. 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. 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. 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, 160 (1982), 49. 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. 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. 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. Google Scholar

[24]

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

[9]

J. Fan and K. Zhao, Global Cauchy problem of $2D$ generalized magnetohydrodynamic equations,, J. Math. Anal. Appl., 420 (2014), 1024. 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. 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. 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. 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. 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. 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. 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. 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. 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, 160 (1982), 49. 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. 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. 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. Google Scholar

[24]

D. H. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics,, SIAM J. Appl. Math., 63 (2003), 1424. 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. doi: 10.1016/j.na.2013.08.024. Google Scholar

[1]

Jishan Fan, Fucai Li, Gen Nakamura. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Conference Publications, 2015, 2015 (special) : 387-394. doi: 10.3934/proc.2015.0387

[2]

Jishan Fan, Fucai Li, Gen Nakamura. A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1757-1766. doi: 10.3934/dcdsb.2018079

[3]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025

[4]

Hyeong-Ohk Bae, Bum Ja Jin. Estimates of the wake for the 3D Oseen equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 1-18. doi: 10.3934/dcdsb.2008.10.1

[5]

Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787

[6]

Ming Lu, Yi Du, Zheng-An Yao. Blow-up phenomena for the 3D compressible MHD equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1835-1855. doi: 10.3934/dcds.2012.32.1835

[7]

Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167

[8]

Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151

[9]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

[10]

Fucai Li, Zhipeng Zhang. Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4279-4304. doi: 10.3934/dcds.2018187

[11]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001

[12]

Thomas Y. Hou, Zuoqiang Shi. Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1449-1463. doi: 10.3934/dcds.2012.32.1449

[13]

Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945

[14]

Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069

[15]

Stefano Scrobogna. Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 5979-6034. doi: 10.3934/dcds.2017259

[16]

Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881

[17]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837

[18]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019164

[19]

Fucai Li, Yanmin Mu. Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1669-1705. doi: 10.3934/dcds.2018069

[20]

Xiaojing Xu, Zhuan Ye. Note on global regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero diffusivity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 585-595. doi: 10.3934/cpaa.2015.14.585

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]