Abstract
In this paper, the time-periodic solution to the compressible viscous quantum magnetohydrodynamic model in a periodic domain is studied. Under the boundedness assumption on the external force, we prove the existence of the time-periodic solution by using the topological degree theory and parabolic regularization method. Furthermore, the uniqueness of the time-periodic solution is shown.
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References
Cai, H., Tan, Z.: Periodic solutions to the compressible magnetohydrodynamic equations in a periodic domain. J. Math. Anal. Appl. 426, 172–193 (2015)
Chen, Y.H., Huang, J.C., Xu, H.Y.: Global stability of large solutions of the \(3\)-D compressible magnetohydrodynamic equations. Nonlinear Anal. Real World Appl. 47, 272–290 (2019)
Fana, J.S., Gao, H.J., Guo, B.L.: Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient. Math. Methods Appl. Sci. 34(17), 2181–2188 (2011)
Galdi, G.P., Kyed, M.: Time-periodic solutions to the Navier-Stokes equations in the three-dimensional whole-space with a non-zero drift term: Asymptotic profile at spatial infinity, Mathematical analysis in fluid mechanics selected recent results, pp. 121–144, Contemp. Math., 710, Amer. Math. Soc., Providence, RI, (2018)
Haas, F.: A magnetohydrodynamic model for quantum plasmas. Phys. Plasmas 12(6), 062117–062126 (2005)
Haas, F.: Quantum Plasmas: An Hydrodynamic Approach. Springer, New York (2011)
Jin, C.H., Yang, T.: Time periodic solution for a 3-D compressible Navier-Stokes system with an external force in \({\mathbb{R}}^{3}\). J. Differ. Equ. 259, 2576–2601 (2015)
Jiang, S., Ju, Q.C., Li, F.C., Xin, Z.P.: Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data. Adv. Math. 259, 384–420 (2014)
Kyed, M.: Existence and regularity of time-periodic solutions to the three-dimensional Navier-Stokes equations in the whole space. Nonlinearity 27, 2909–2935 (2014)
Li, H.Y., Cheng, M., Yan, W.P.: Global existence and large time behavior of solutions for compressible quantum magnetohydrodynamics flows in \({\mathbb{T}}^{3}\). J. Math. Anal. Appl. 452, 1209–1228 (2017)
Lu, L., Huang, B.: On local strong solutions to the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with vacuum and zero heat conduction. Nonlinear Anal. Real World Appl. 31, 409–430 (2016)
Li, Y., Sun, Y.H.: Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids. J. Differ. Equ. 267, 3827–3851 (2019)
Li, Y.P., Yang, X.F.: Stability of stationary solution for the compressible viscous magnetohydrodynamic equations with large potential force in bounded domain. J. Differ. Equ. 262, 3169–3193 (2017)
Ou, Y.B., Yang, L.: Incompressible limit of non-isentropic compressible magnetohydrodynamic equations with zero magnetic diffusivity in bounded domains. Nonlinear Anal. Real World Appl. 49, 1–23 (2019)
Pu, X. K., Xu, X.L.: Decay rates of the magnetohydrodynamic model for quantum plasmas, Z. Angew. Math. Phys., 68(18), 17 pp (2017)
Tsuda, K.: Existence and Stability of Time Periodic Solution to the Compressible Navier-Stokes-Korteweg System on \({\mathbb{R}}^{3}\). J. Math. Fluid Mech. 18, 157–185 (2016)
Tan, Z., Tong, L.L., Wang, Y.: Large time behavior of the compressible magnetohydrodynamic equations with Coulomb force. J. Math. Anal. Appl. 427, 600–617 (2015)
Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)
Xi, X.Y.: Decay rates of the compressible quantum magnetohydrodynamic model. J. Math. Anal. Appl. 475, 403–422 (2019)
Xi, X.Y., Pu, X.K., Guo, B.L.: Long-time behavior of solutions for the compressible quantum magnetohydrodynamic model in \({\mathbb{R}}^{3}\). Z. Angew. Math. Phys. 70(7), 16 (2019)
Yang, J.W., Ju, Q.C.: Global existence of the three-dimensional viscous quantum magnetohydrodynamic model. J. Math. Phys. 55(8), 12 (2014)
Yang, J.W.: Low Mach number limit of the viscous quantum magnetohydrodynamic model. J. Math. Anal. Appl. 455, 1110–1123 (2017)
Yang, Y., Tao, Q.: Time-periodic solution to the compressible nematic liquid crystal flows in periodic domain. Math. Methods Appl. Sci. 41(1), 28–45 (2018)
Yan, W.P.: Large data existence result for the steady full compressible magnetohydrodynamic flows in three dimensional. Ann. Mat. Pura Appl. 195(4), 1–27 (2016)
Acknowledgements
The authors would like to thank the anonymous referees sincerely for their valuable suggestions and comments which greatly improve the exposition of the paper. Yang is partially supported by the Guangdong Basic and Applied Basic Research Foundation (No.2020A1515010446), the National Science Foundation of China (No.11971320, No.11671155 and No.11701384) and China Scholarship Council (No.201908440614). Tao is partially supported by the Guangdong Basic and Applied Basic Research Foundation (No.2020A1515010530) and China Scholarship Council (No.201908440027).
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Yang, Y., Zhou, Y. & Tao, Q. Time-periodic solution to the compressible viscous quantum magnetohydrodynamic model. Z. Angew. Math. Phys. 71, 103 (2020). https://doi.org/10.1007/s00033-020-01328-2
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DOI: https://doi.org/10.1007/s00033-020-01328-2