Skip to main content
SpringerLink
Log in

Local Well-Posedness of Strong Solutions for the Nonhomogeneous MHD Equations with a Slip Boundary Conditions

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

This article is concerned with the 3D nonhomogeneous incompressible magneto- hydrodynamics equations with a slip boundary conditions in bounded domain. We obtain weighted estimates of the velocity and magnetic field, and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agapito R, Schonbek M. Non-uniform decay of MHD equations with and without magnetic diffusion. Commu Part Diff Eq, 2007, 32(11): 1791–1821

    Article  MathSciNet  MATH  Google Scholar 

  2. Beirão da Veiga H, Valli A. On the Euler equations for nonhomogeneous fluids II. J Math Anal Appl, 1980, 73(2): 338–350

    Article  MathSciNet  MATH  Google Scholar 

  3. Berselli L, Spirito S. On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains. Commun Math Phys, 2012, 316(1): 171–198

    Article  MathSciNet  MATH  Google Scholar 

  4. Biskamp D. Nonlinear magnetohydrodynamics. Cambridge University Press, 1997

    Google Scholar 

  5. Chen G Q, W ang D H. Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z Angew Math Phys, 2003, 54(4): 608–632

    Article  MathSciNet  Google Scholar 

  6. Chen P F, Xiao Y L, Zhang H. Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equation with special slip boundary conditions. Electron J Differ Eq, 2017, 2017(169): 1–13

    MathSciNet  MATH  Google Scholar 

  7. Chen Q, Tan Z, Wang Y J. Strong solutions to the incompressible magnetohydrodynamic equations. Math Method Appl Sci, 2011, 34(1): 94–107

    Article  MathSciNet  MATH  Google Scholar 

  8. Ducomet B, Feireisl E. The equations of Magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars. Commun Math Phys, 2006, 266(3): 595–629

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan J S, Jiang S, Nakamura G. Vanishing shear viscosity limit in the magnetohydrodynamic equations. Commun Math Phys, 2007, 270(3): 691–708

    Article  MathSciNet  MATH  Google Scholar 

  10. Feireisl E, Novotný A, Petzeltová H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3(4): 358–392

    Article  MathSciNet  MATH  Google Scholar 

  11. Ferreira L C F, Planas G, Vllamizar-Roa E J. On the Nonhomogeneous Navier-Stokes System with Navier Friction Boundary Conditions. SIAM J Math Anal, 2013, 45(4): 2576–2595

    Article  MathSciNet  MATH  Google Scholar 

  12. Freidberg J. Ideal Magnetohydrodynamics. New York London Plenum Press, 1987

    Book  Google Scholar 

  13. Guo B L, Wang G W. Vanishing viscosity limit for the 3D magnetohydrodynamic system with generalized Navier slip boundary conditions. Math Meth Appl Sci, 2016, 39(15): 4526–4534

    Article  MathSciNet  MATH  Google Scholar 

  14. Hoff D, Tsyganov E. Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z Angew Math Phys, 2005, 56(5): 791–804

    Article  MathSciNet  MATH  Google Scholar 

  15. Hu X P, Wang D H. Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch Rational Mech Anal, 2010, 197(1): 203–238

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang X D, Wang Y. Global strong solution to the 2D nonhomogeneous incompressible MHD system. J Differ Equations, 2013, 254(2): 511–527

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang X D, Wang Y. Global strong solution of 3D inhomogeneous Navier-Stokes equations with densitydependent viscosity. J Differ Equations, 2015, 259(4): 1606–1627

    Article  MATH  Google Scholar 

  18. Jiu Q S, Wang, Xin Z P. Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces. J Differ Equations, 255(3): 351–404

  19. Ladyzhenskaya O A. The Mathematical Theory of Viscous Incompressible Flow. second English edition, Mathematics and Its Applications. Vol. 2. New York, London, Paris: Gordon and Breach, Science Publishers, 1969

    Google Scholar 

  20. Ladyzhenskaya O A, Solonnikov V A. Unique solvability of an initial and boundary value problem for viscous incompressible non-homogeneous fluids. J Sov Math, 1978, 9(5): 697–749

    Article  MATH  Google Scholar 

  21. Li F C, Zhang Z P. Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class. Discrete and continuous dynamical systems, 2018, 38(9): 4279–4304

    Article  MathSciNet  MATH  Google Scholar 

  22. Li H Y. Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity. Math Methods Appl Sci, 2018, 41(8): 3062–3092

    Article  MathSciNet  MATH  Google Scholar 

  23. Li J. Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density. J Differ Equations, 2017, 263(10): 6512–6536

    Article  MathSciNet  MATH  Google Scholar 

  24. Lv B Q, Xu Z H, Zhong X. Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum. J Math Pures Appl, 2017, 108(1): 41–62

    Article  MathSciNet  MATH  Google Scholar 

  25. Paicu M, Zhang P, Zhang Z F. Global unique solvability of homogeneous Navier-Stokes equations with bounded density. Commun Part Diff Eq, 2013, 38(7): 1208–1234

    Article  MATH  Google Scholar 

  26. Sart R. Existence of finite energy weak solutions for the equations MHD of compressible fluids. Appl Anal, 2009, 88(3): 357–379

    Article  MathSciNet  MATH  Google Scholar 

  27. Simon J. Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J Math Anal, 1990, 21(5): 1093–1117

    Article  MathSciNet  MATH  Google Scholar 

  28. Temam R. Navier-Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: SIAM, 1983

    Google Scholar 

  29. Wu H W. Strong solutions to the incompressible magnetohydrodynamic equations with vacuum. Comput Math Appl, 2011, 61(9): 2742–2753

    Article  MathSciNet  MATH  Google Scholar 

  30. Wu J H. Generalized MHD equations. J Differ Equations, 2003, 195(2): 284–312

    Article  MathSciNet  MATH  Google Scholar 

  31. Xiao Y L, Xin Z P. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Commun Pur Appl Math, 2007, 60(7): 1027–1055

    Article  MathSciNet  MATH  Google Scholar 

  32. Xiao Y L, Xin Z P, Wu J H. Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition. J Funct Anal, 2009, 257(11): 3375–3394

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu L, Zhang P. Global small solutions to three-dimensional incompressible magnetohydrodynamical system. SIAM J Math Anal, 2015, 47(1): 26–65

    Article  MathSciNet  MATH  Google Scholar 

  34. Yu H B, Zhang P X, Shi X J. Global strong solutions to the 3D incompressible MHD equations with density-dependent viscosity. Comput Math Appl, 2018, 75(8): 2825–2834

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhumadian Academy of Industry Innovation and Development, Huanghuai University, Zhumadian, 463000, China

    Hongmin Li

  2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Yuelong Xiao

Authors
  1. Hongmin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuelong Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuelong Xiao.

Additional information

This work was supported by Natural Science Foundation of China (11871412).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Xiao, Y. Local Well-Posedness of Strong Solutions for the Nonhomogeneous MHD Equations with a Slip Boundary Conditions. Acta Math Sci 40, 442–456 (2020). https://doi.org/10.1007/s10473-020-0210-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-020-0210-x

Key words

2010 MR Subject Classification

Search

Navigation