Skip to main content
SpringerLink
Log in

LOCAL SOLVABILITY OF FREE BOUNDARY PROBLEMS IN IDEAL COMPRESSIBLE MAGNETOHYDRODYNAMICS WITH AND WITHOUT SURFACE TENSION

  • Published:
Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

Results are presented on local-in-time solvability of free boundary problems for a system of ideal compressible magnetohydrodynamics. A free plasma-vacuum interface problem and a free boundary problem with boundary conditions on a contact discontinuity are considered. An approach is given for proving the local existence and uniqueness of smooth solutions of these problems with and without surface tension.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

REFERENCES

  1. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Vol. 8: Electrodynamics of Continuous Media (Pergamon Press, 1960).

    Google Scholar 

  2. I. B. Bernstein, E. A. Frieman, M. D. Kruskal, and R. M. Kulsrud, “An Energy Principle for Hydromagnetic Stability Problems," Proc. Roy. Soc. London. Ser. A. 244, 17–40 (1958).

    ADS  MathSciNet  MATH  Google Scholar 

  3. Y. Trakhinin, “On the Well-Posedness of a Linearized Plasma — Vacuum Interface Problem in Ideal Compressible MHD," J. Different. Equat. 249, 2577–2599 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  4. J. P. Goedbloed, R. Keppens, and S. Poedts, Advanced Magnetohydrodynamics: with Applications to Laboratory and Astrophysical Plasmas (Cambridge Univ. Press, Cambridge, 2010).

    Book  Google Scholar 

  5. H. Lindblad, “Well-Posedness for the Motion of a Compressible Liquid with Free Surface Boundary," Comm. Math. Phys. 260, 319–392 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  6. Y. Trakhinin, “Local Existence for the Free Boundary Problem for Nonrelativistic and Relativistic Compressible Euler Equations with a Vacuum Boundary Condition," Comm. Pure Appl. Math. 62, 1551–1594 (2009).

    Article  MathSciNet  Google Scholar 

  7. R. Samulyak, J. Du, J. Glimm, and Z. Xu, “A Numerical Algorithm for MHD of Free Surface Flows at Low Magnetic Reynolds Numbers," J. Comput. Phys. 226, 1532–1549 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  8. F. Yang, A. Khodak, and H. A. Stone, “The Effects of a Horizontal Magnetic Field on the Rayleigh — Taylor Instability," Nuclear Materials Energy 18, 175–181 (2019).

    Article  Google Scholar 

  9. J. M. Stone and T. Gardiner, “Nonlinear Evolution of the Magnetohydrodynamic Rayleigh — Taylor Instability," Phys. Fluids 19, 094104 (2007).

    Article  ADS  Google Scholar 

  10. J. M. Stone and T. Gardiner, “The Magnetic Rayleigh — Taylor Instability in Three Dimensions," Astrophys. J. 671, 1726–1735 (2007).

    Article  ADS  Google Scholar 

  11. A. M. Blokhin and Yu. L. Trakhinin, Stability of Strong Discontinuities in Magnetohydrodynamics and Electrohydrodynamics (Nova Science Pub Inc, 2003).

    Google Scholar 

  12. Y. Trakhinin, “On Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis," Arch. Ration. Mech. Anal. 177, 331–366 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  13. Y. Trakhinin, “The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics," Arch. Ration. Mech. Anal. 191, 245–310 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  14. J. M. Delhaye, “Jump Conditions and Entropy Sources in Two-Phase Systems. Local Instant Formulation," Intern. J. Multiphase Flow 1, 395–409 (1974).

    Article  Google Scholar 

  15. M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow (Springer, N. Y., 2011).

    Book  Google Scholar 

  16. Y. Trakhinin and T. Wang, Nonlinear Stability of MHD Contact Discontinuities with Surface Tension [Website]. https://arxiv.org/pdf/2105.04977.pdf.

  17. V. I. Nalimov, “Cauchy — Poisson Problems", in Continuum Dynamics (Akad. Nauk SSSR. Sib Otd. Inst. Gidrodinamiki, 1974) [in Russian].

  18. V. I. Nalimov and V. V. Pukhnachev, Unsteady Motion of an Ideal Fluid with a Free Boundary (Novosibirsk State University, Novosibirsk, 1975) [in Russian].

  19. V. I. Nalimov, “Justification of Approximate Models of the Theory of Plane Unsteady Waves," in: Nonlinear Problems of the Theory of Surface and Internal Waves (Nauka. Sibirskoe Otdelenie, Novosibirsk, 1985) [in Russian].

  20. P. I. Plotnikov, “Ill-Posedness of the Nonlinear Problem of the Development of the Rayleigh — Taylor Instability," Zapiski Nauchnykh Seminarov 96, 240–296 (1980).

    MathSciNet  MATH  Google Scholar 

  21. Y. Guo and I. Tice, “Compressible, Inviscid Rayleigh — Taylor Instability," Indiana Univ. Math. J. 60, 677–711 (2011).

    Article  MathSciNet  Google Scholar 

  22. Y. Trakhinin, “On Well-Posedness of the Plasma — Vacuum Interface Problem: The Case of Non-Elliptic Interface Symbol," Comm. Pure Appl. Anal. 15, 1371–1399 (2016).

    Article  MathSciNet  Google Scholar 

  23. D. Ebin, “The Equations of Motion of a Perfect Fluid with Free Boundary Are Not Well-Posed," Comm. Part. Different. Equat. 12, 1175–1201 (1987).

    Article  MathSciNet  Google Scholar 

  24. P. Secchi and Y. Trakhinin, “Well-posedness of the Plasma — Vacuum Interface Problem," Nonlinearity 27, 105–169 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  25. Y. Trakhinin and T. Wang, “Well-Posedness of Free Boundary Problem in Non-Relativistic and Relativistic Ideal Compressible Magnetohydrodynamics," Arch. Ration. Mech. Anal. 239, 1131–1176 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  26. L. Hörmander, “The Boundary Problems of Physical Geodesy," Arch. Ration. Mech. Anal. 62, 1–52 (1976).

    Article  ADS  MathSciNet  Google Scholar 

  27. S. Alinhac, “Existence d’ondes de raréfaction pour des systèmes quasilinéaires hyperboliques multidimensionnels," Comm. Part. Different. Equat. 14, 173–230 (1989).

    Article  MathSciNet  Google Scholar 

  28. P. D. Lax and R. S. Phillips, “Local Boundary Conditions for Dissipative Symmetric Linear Differential Operators," Comm. Pure Appl. Math. 13, 427–455 (1960).

    Article  MathSciNet  Google Scholar 

  29. P. Secchi and Y. Trakhinin, “Well-posedness of the Linearized Plasma — Vacuum Interface Problem," Interfaces Free Bound 15, 323–357 (2013).

    Article  MathSciNet  Google Scholar 

  30. Y. Trakhinin, “Stability of Relativistic Plasma — Vacuum Interfaces," J. Hyperbolic Different. Equat. 9, 469–509 (2012).

    Article  MathSciNet  Google Scholar 

  31. A. Morando, Y. Trakhinin, and P. Trebeschi, “Well-Posedness of the Linearized Problem for MHD Contact Discontinuities," J. Different. Equat. 258, 2531–2571 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  32. A. Morando, Y. Trakhinin, and P. Trebeschi, “Local Existence of MHD Contact Discontinuities," Arch. Ration. Mech. Anal. 228, 691–742 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  33. Y. Trakhinin and T. Wang, “Well-Posedness for the Free-Boundary Ideal Compressible Magnetohydrodynamic Equations with Surface Tension," Math. Ann. (2021). https://doi.org/10.1007/s00208-021-02180-z.

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Yu. L. Trakhinin

Authors
  1. Yu. L. Trakhinin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. L. Trakhinin.

Additional information

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, 2021, Vol. 62, No. 4, pp. 181-190. https://doi.org/10.15372/PMTF20210418.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Trakhinin, Y.L. LOCAL SOLVABILITY OF FREE BOUNDARY PROBLEMS IN IDEAL COMPRESSIBLE MAGNETOHYDRODYNAMICS WITH AND WITHOUT SURFACE TENSION. J Appl Mech Tech Phy 62, 684–691 (2021). https://doi.org/10.1134/S0021894421040180

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0021894421040180

Keywords

Search

Navigation