Skip to main content
SpringerLink
Log in

Local well-posedness of the free-boundary problem in compressible resistive magnetohydrodynamics

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh–Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand–Shkoller (J Am Math Soc 20(3):829–930, 2007, Arch Ration Mech Anal 206(2):515–616, 2012) to construct the approximation solutions. One of the key observations is that the Christodoulou–Lindblad type elliptic estimate (Christodoulou and Lindblad in Commun Pure Appl Math 53(12):1536–1602, 2000) together with magnetic diffusion not only gives the common control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.

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. Alazard, T., Burq, N., Zuily, C.: On the Cauchy problem for gravity water waves. Inventiones Mathematicae 198(1), 71–163 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Alazard, T., Delort, J.-M.: Global solutions and asymptotic behavior for two dimensional gravity water waves. Annales Scientifiques de l’École Normale Supérieure 48(5), 1149–1238 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Alinhac, S.: Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems. Commun. Part. Differ. Equ. 14(2), 173–230 (1989)

    MATH  Google Scholar 

  4. Chen, P., Ding, S.: Inviscid limit for the free-boundary problems of MHD equations with or without surface tension, preprint (2019). arXiv: 1905.13047

  5. Chen, S.-X.: Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary. Translated from Chin. Ann. Math. 3(2), 222–232 (1982). Front. Math. China 2(1), 87–102 (2007)

  6. Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Commun. Pure Appl. Math. 53(12), 1536–1602 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Coutand, D., Hole, J., Shkoller, S.: Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit. SIAM J. Math. Anal. 45(6), 3690–3767 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Coutand, D., Shkoller, S.: A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete Contin. Dyn. Syst. (Ser. S) 3(3), 429–449 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. 206(2), 515–616 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Coutand, D., Lindblad, H., Shkoller, S.: A priori estimtes for the free-boundary 3D compressible Euler equations in physical vacuum. Commun. Math. Phys. 296(2), 559–587 (2007)

    MATH  Google Scholar 

  12. Deng, Y., Ionescu, A.D., Pausader, B., Pusateri, F.: Global solutions of the gravity-capillary water-wave system in three dimensions. Acta Math. 219(2), 213–402 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Disconzi, M.M., Kukavica, I.: A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolut. Equ. Control Theory 8(3), 503–542 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Disconzi, M.M., Luo, C.: On the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Arch. Ration. Mech. Anal. 237, 829–897 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Ebin, D.G.: The equations of motion of a perfect fluid with free boundary are not well posed. Commun. Part. Differ. Equ. 12(10), 1175–1201 (1987)

    MathSciNet  MATH  Google Scholar 

  16. Germain, M., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. 175(2), 691–754 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Germain, M., Masmoudi, N., Shatah, J.: Global existence for capillary water waves. Commun. Pure Appl. Math. 68(4), 625–687 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Ginsberg, D., Lindblad, H., Luo, C.: Local well-posedness for the motion of a compressible, self-gravitating liquid with free surface boundary. Arch. Rational Mech. Anal. 236(2), 603–733 (2020)

    MathSciNet  MATH  Google Scholar 

  19. Goedbloed, H., Keppens, R., Poedts, S.: Magnetohydrodynamics of Laboratory and Astrophysical plasmas. Cambridge University Press (2019)

    MATH  Google Scholar 

  20. Gu, X.: Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition. Commun. Pure Appl. Anal. 18(2), 569–602 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Gu, X.: Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the Rayleigh–Taylor sign condition, preprint (2017). arXiv: 1712.02152

  22. Gu, X., Wang, Y.: On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations. Journal de Mathématiques Pures et Appliquées 128, 1–41 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Guo, B., Zeng, L., Ni, G.: Decay rates for the viscous incompressible MHD with and without surface tension. Comput. Math. Appl. 77(12), 3224–3249 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Hao, C., Luo, T.: A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows. Arch. Ration. Mech. Anal. 212(3), 805–847 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Hao, C., Luo, T.: Ill-posedness of free boundary problem of the incompressible ideal MHD. Commun. Math. Phys. 376, 259–286 (2020)

    MathSciNet  MATH  Google Scholar 

  26. Hao, C., Luo, T.: Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations. J. Differ. Equ. 299, 542–601 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Ifrim, M., Tataru, D.: Two dimensional water waves in holomorphic coordinates II: global solutions. Bull. Soc. Math. France 144(2), 369–394 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Jang, J., Masmoudi, N.: Well-posedness of compressible Euler equations in a physical vacuum. Commun. Pure Appl. Math. 68(1), 61–111 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18(3), 605–654 (2005)

    MathSciNet  MATH  Google Scholar 

  30. Lee, D.: Initial value problem for the free boundary magnetohydrodynamics with zero magnetic boundary condition. Commun. Math. Sci. 16(3), 589–615 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Lee, D.: Uniform estimate of viscous free-boundary magnetohydrodynamics with zero vacuum magnetic field. SIAM J. Math. Anal. 49(4), 2710–2789 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Lindblad, H.: Well-posedness for the linearized motion of an incompressible liquid with free surface boundary. Commun. Pure Appl. Math. 56, 153–197 (2003)

    MathSciNet  MATH  Google Scholar 

  33. Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162(1), 109–194 (2005)

    MathSciNet  MATH  Google Scholar 

  34. Lindblad, H.: Well-posedness for the linearized motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 236, 281–310 (2003)

    MathSciNet  MATH  Google Scholar 

  35. Lindblad, H.: Well-posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260(2), 319–392 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Lindblad, H., Luo, C.: A priori estimates for the compressible Euler equations for a liquid with free surface boundary and the incompressible limit. Commun. Pure Appl. Math. 71(7), 1273–1333 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Luo, C.: On the motion of a compressible gravity water wave with vorticity. Ann. PDE 4(2), 2506–2576 (2018)

    MathSciNet  Google Scholar 

  38. Luo, C., Zhang, J.: A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary. Nonlinearity 33(4), 1499–1527 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Luo, C., Zhang, J.: A priori estimates for the incompressible free-boundary magnetohydrodynamics equations with surface tension. SIAM J. Math. Anal. 53, 2595–2630 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Luo, C., Zhang, J.: Local well-posedness for the motion of a compressible gravity water wave with vorticity, Preprint (2020). J. Differ. Equ. 332, 333–403 (2022)

    MATH  Google Scholar 

  41. Luo, T., Xin, Z., Zeng, H.: Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation. Arch. Ration. Mech. Anal. 213(3), 763–831 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Masmoudi, N., Rousset, F.: Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations. Arch. Ration. Mech. Anal. 223(1), 301–417 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Morando, A., Trakhinin, Y., Trebeschi, P.: Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD. Q. Appl. Math. 72(3), 549–587 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Ohno, M., Shirota, T.: On the initial-boundary-value problem for the linearized equations of magnetohydrodynamics. Arch. Ration. Mech. Anal. 144, 259–299 (1998)

    MathSciNet  MATH  Google Scholar 

  45. Padula, M., Solonnikov, V.A. : On the free boundary problem of magnetohydrodynamics. Zap. Nauchn. Semin. POMI 385, 135–186 (2010). J. Math. Sci. (N.Y.) 178(3), 313–344 (2011)

  46. Secchi, P.: Well-posedness for mixed problems for the equations of ideal magneto-hydrodynamics. Archiv. der. Math. 64(1995), 237–245 (1995)

    MathSciNet  MATH  Google Scholar 

  47. Secchi, P.: Well-posedness of characteristic symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 134, 155–197 (1996)

    MathSciNet  MATH  Google Scholar 

  48. Secchi, P., Trakhinin, Y.: Well-posedness of the plasma-vacuum interface problem. Nonlinearity 27(1), 105–169 (2013)

    MathSciNet  MATH  Google Scholar 

  49. Shatah, J., Zeng, C.: Geometry and a priori estimates for free boundary problems of the Euler’s equation. Commun. Pure Appl. Math. 61(5), 698–744 (2008)

    MathSciNet  MATH  Google Scholar 

  50. Shatah, J., Zeng, C.: A priori estimates for fluid interface problems. Commun. Pure Appl. Math. 61(6), 848–876 (2008)

    MathSciNet  MATH  Google Scholar 

  51. Shatah, J., Zeng, C.: Local well-posedness for fluid interface problems. Arch. Ration. Mech. Anal. 199(2), 653–705 (2011)

    MathSciNet  MATH  Google Scholar 

  52. Sun, Y., Wang, W., Zhang, Z.: Well-posedness of the plasma-vacuum interface problem for ideal incompressible MHD. Arch. Ration. Mech. Anal. 234(1), 81–113 (2019)

    MathSciNet  MATH  Google Scholar 

  53. Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis, vol. 106. American Mathematical Society (2006)

    MATH  Google Scholar 

  54. Taylor, M.: Partial Differential Equations I: Basic Theory. Applied Mathematical Sciences, vol. 115. Springer, New York (2011)

    MATH  Google Scholar 

  55. Trakhinin, Y.: Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition. Commun. Pure Appl. Math. 62(11), 1551–1594 (2009)

    MathSciNet  MATH  Google Scholar 

  56. Trakhinin, Y.: On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol. Commun. Pure Appl. Anal. 15(4), 1371–1399 (2016)

    MathSciNet  MATH  Google Scholar 

  57. Trakhinin, Y., Wang, T.: Well-posedness of free boundary problem in non-relativistic and relativistic ideal compressible magnetohydrodynamics. Arch. Rational Mech. Anal. 239(2), 1131–1176 (2021)

    MathSciNet  MATH  Google Scholar 

  58. Wang, Y., Xin, Z.: Vanishing viscosity and surface tension limits of incompressible viscous surface waves. SIAM J. Math. Anal. 53(1), 574–648 (2021)

    MathSciNet  MATH  Google Scholar 

  59. Wang, Y., Xin, Z.: Global well-posedness of free interface problems for the incompressible inviscid resistive MHD. Commun. Math. Phys. 388(3), 1323–1401 (2021)

    MathSciNet  MATH  Google Scholar 

  60. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997)

    MathSciNet  MATH  Google Scholar 

  61. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999)

    MathSciNet  MATH  Google Scholar 

  62. Wu, S.: Almost global well-posedness of the 2-D full water wave problem. Invent. Math. 177(1), 45–135 (2009)

    MathSciNet  Google Scholar 

  63. Wu, S.: Global well-posedness of the 3-D full water wave problem. Invent. Math. 184(1), 125–220 (2011)

    MathSciNet  MATH  Google Scholar 

  64. Yanagisawa, T., Matsumura, A.: The fixed boundary value problems for the equations of ideal magnetohydrodynamics with a perfectly conducting wall condition. Commun. Math. Phys. 136(1), 119–140 (1991)

    MathSciNet  MATH  Google Scholar 

  65. Zhang. J.: A priori estimates for the free-boundary problem of compressible resistive MHD equations and incompressible limit, preprint (2019). arxiv: 1911.04928

  66. Zhang, P., Zhang, Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61(7), 877–940 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Junyan Zhang

Authors
  1. Junyan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junyan Zhang.

Additional information

Communicated by F.-H. Lin.

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 (e.g. a society or other partner) 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

Zhang, J. Local well-posedness of the free-boundary problem in compressible resistive magnetohydrodynamics. Calc. Var. 62, 124 (2023). https://doi.org/10.1007/s00526-023-02462-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-023-02462-1

Mathematics Subject Classification

Search

Navigation