Skip to main content
SpringerLink
Log in

Global well-posedness of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with large viscosity

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, the initial-boundary value problem for the two-dimensional viscous, compressible and heat-conducting magnetohydrodynamics (MHD) equations with vacuum is considered. We show that the strong solution exists globally in the time provided the viscosity coefficient \(\mu \) is suitably large, and there is no small restriction on the initial data. As a result, we extend the works given by Li and Shang [20] for the full compressible Navier–Stokes equations and by Liu [25] for the isentropic compressible MHD equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Agmon, Shmuel, Douglis, Avron, Nirenberg, Louis: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, Gui-Qiang., Wang, Dehua: Global solutions of nonlinear magnetohydrodynamics with large initial data. J. Differ. Equ. 182(2), 344–376 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Yazhou, Huang, Bin, Shi, Xiaoding: Global strong and weak solutions to the initial-boundary-value problem of two-dimensional compressible MHD system with large initial data and vacuum. SIAM J. Math. Anal. 54(3), 3817–3847 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cho, Yonggeun, Kim, Hyunseok: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228(2), 377–411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deng, Xuemei, Zhang, Peixin, Zhao, Junning: Global classical solution to the three-dimensional isentropic compressible Navier-Stokes equations with general initial data. Sci. China Math. 55(12), 2457–2468 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ducomet, Bernard, Feireisl, Eduard: The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars. Comm. Math. Phys. 266(3), 595–629 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, Jishan, Jiang, Song, Nakamura, Gen: Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data. J. Differ. Equ. 251(8), 2025–2036 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan, Jishan, Wanghui, Yu.: Strong solution to the compressible magnetohydrodynamic equations with vacuum. Nonlinear Anal. Real World Appl. 10(1), 392–409 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan, Xinyu, Li, Jiaxu, Li, Jing: Global existence of strong and weak solutions to 2D compressible Navier-Stokes system in bounded domains with large data and vacuum. Arch. Ration. Mech. Anal. 245(1), 239–278 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hao, Tiantian: Global solutions of 3-D inhomogeneous Navier-Stokes system with large viscosity in one variable. Commun. Math. Res. 38(1), 62–80 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hou, X., Jiang, M., Peng, H.: Global strong solution to 3D full compressible magnetohydrodynamic flows with vacuum at infinity. Z. Angew. Math. Phys. 73(1), 1–18 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xianpeng, Hu., Wang, Dehua: Global solutions to the three-dimensional full compressible magnetohydrodynamic flows. Comm. Math. Phys. 283(1), 255–284 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yuxi, Hu., Qiangchang, Ju.: Global large solutions of magnetohydrodynamics with temperature-dependent heat conductivity. Z. Angew. Math. Phys. 66(3), 865–889 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, Xiangdi, Li, Jing: Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations. Arch. Ration. Mech. Anal. 227(3), 995–1059 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, Xiangdi, Li, Jing, Xin, Zhouping: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Comm. Pure Appl. Math. 65(4), 549–585 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jenssen, Helge Kristian, Karper, Trygve Klovning: One-dimensional compressible flow with temperature dependent transport coefficients. SIAM J. Math. Anal. 42(2), 904–930 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kawashima, Shuichi: Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics. Japan J. Appl. Math. 1(1), 207–222 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kawashima, Shuichi, Okada, Mari: Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A Math. Sci. 58(9), 384–387 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kawohl, Bernhard: Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas. J. Differential Equations 58(1), 76–103 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, H., Shang, Z.: Global strong solution to the two-dimensional full compressible Navier-Stokes equations with large viscosity. J. Math. Fluid Mech. 24(1), 1–29 (2022)

    MathSciNet  MATH  Google Scholar 

  21. Li, Yachun, Shang, Zhaoyang: Global large solutions to planar magnetohydrodynamics equations with temperature-dependent coefficients. J. Hyperbolic Differ. Equ. 16(3), 443–493 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, Yang, Sun, Yongzhong: On global-in-time weak solutions to a two-dimensional full compressible nonResistive MHD system. SIAM J. Math. Anal. 53(4), 4142–4177 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liang, Zhilei: Global strong solutions of Navier-Stokes equations for heat-conducting compressible fluids with vacuum at infinity. J. Math. Fluid Mech. 23(1), 1–22 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, Hongxia, Yang, Tong, Zhao, Huijiang, Zou, Qingyang: One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data. SIAM J. Math. Anal. 46(3), 2185–2228 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, Yang: Global classical solutions of 3D isentropic compressible MHD with general initial data. Z. Angew. Math. Phys. 66(4), 1777–1797 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, Y., Zhong, X.: Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum. Z. Angew. Math. Phys. 71(6), 1–25 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, Yanlin, Zhang, Ping: Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient. J. Funct. Anal. 279(10), 108736, 33 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, Yanlin, Zhang, Ping: Remark on 3-D Navier-Stokes system with strong dissipation in one direction. Commun. Pure Appl. Anal. 20(7–8), 2765–2787 (2021)

    Article  MathSciNet  Google Scholar 

  29. Paicu, Marius, Zhang, Ping: Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction. Sci. China Math. 62(6), 1175–1204 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pan, Ronghua, Zhang, Weizhe: Compressible Navier-Stokes equations with temperature dependent heat conductivity. Commun. Math. Sci. 13(2), 401–425 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xueke, Pu., Guo, Boling: Global existence and convergence rates of smooth solutions for the full compressible MHD equations. Z. Angew. Math. Phys. 64(3), 519–538 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shang, Zhaoyang: Global strong solutions to the two-dimensional full compressible Navier-Stokes equations with vacuum. J. Math. Anal. Appl. 481(1), 123459, 27 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shang, Zhaoyang: Global large solutions to the Cauchy problem of planar magnetohydrodynamics equations with temperature-dependent coefficients. J. Dyn. Control Syst. 28(1), 163–205 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tan, Zhong, Yang, Tong, Zhao, Huijiang, Zou, Qingyang: Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data. SIAM J. Math. Anal. 45(2), 547–571 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tang, Tong, Gao, Hongjun: Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete Contin. Dyn. Syst. Ser. B 21(5), 1617–1633 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vaigant, V.A., Kazhikhov, A.V.: On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid. Sibirsk. Mat. Zh. 36(6), 1283–1316, ii (1995)

    MathSciNet  Google Scholar 

  37. Wen, Huanyao, Zhu, Changjiang: Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data. SIAM J. Math. Anal. 49(1), 162–221 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ye, Yulin: Global classical solution to the Cauchy problem of the 3-D compressible Navier-Stokes equations with density-dependent viscosity. Acta Math. Sci. Ser. B (Engl. Ed.) 36(5), 1419–1432 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Haibo, Yu., Zhang, Peixin: Global strong solutions to the 3D full compressible Navier-Stokes equations with density-temperature-dependent viscosities in bounded domains. J. Differ. Equ. 268(12), 7286–7310 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Haibo, Yu., Zhao, Junning: Global classical solutions to the 3D isentropic compressible Navier-Stokes equations in a bounded domain. Nonlinearity 30(1), 361–381 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, Peixin, Zhao, Junning: Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data. Sci. China Math. 57(7), 1463–1478 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhong, Xin: Strong solutions to the Cauchy problem of two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction. J. Differ. Equ. 268(9), 4921–4944 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zlotnik, Alexander A.: Uniform estimates and the stabilization of symmetric solutions of a system of quasilinear equations. Differ. Uravn. 36(5), 634–646, 718 (2000)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported partially by Chinese National Natural Science Foundation under grant 11831011 and by China Postdoctoral Science Foundation under grant 2021M692089. The authors would like to thank the anonymous referees for their helpful comments, which improve the presentation of the paper.

Author information

Authors and Affiliations

  1. School of Finance, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, People’s Republic of China

    Wei Li & Zhaoyang Shang

  2. Postdoctoral Station of Applied Economics, Fudan University, Shanghai, 200433, People’s Republic of China

    Wei Li

  3. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Zhaoyang Shang

Authors
  1. Wei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhaoyang Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaoyang Shang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, W., Shang, Z. Global well-posedness of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with large viscosity. Z. Angew. Math. Phys. 73, 205 (2022). https://doi.org/10.1007/s00033-022-01837-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01837-2

Keywords

Mathematics Subject Classification

Search

Navigation