Skip to main content
SpringerLink
Log in

A Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536–1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.

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

Abbreviations

x = (x i):

Eulerian coordinates

y = (y a):

Lagrangian coordinates

∂ :

Spatial derivative in x

∇:

Covariant derivative in y

v :

The velocity field in Eulerian coordinates

u :

The velocity field in Lagrangian coordinates

B :

The magnetic field in Eulerian coordinates

β :

The magnetic field in Lagrangian coordinates

p :

Fluid pressure

\({P = p + \frac{1}{8\pi}|B|^{2}}\) :

Total pressure

D t :

=\({\frac{\partial}{\partial t}\left|_{y={\rm const}} = \frac{\partial}{\partial t}\right|_{x={\rm const}} + v^k\frac{\partial}{\partial x^k}}\)

g :

The Riemannian metric defined by \({g_{ab} = \sum_{i, j}\delta_{ij}\frac{\partial x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b}}\)

γ :

The induced metric on the tangent space of the boundary which can be extended to be 0 on the orthogonal complement of the tangent space of the boundary. Also, it can be extended to be a pseudo-Riemannian metric in the whole domain

Π :

Orthogonal projection to the tangent space of the boundary

θ :

The second fundamental form of the boundary

\({\iota_0}\) :

The injectivity radius of the normal exponential map

References

  1. Ambrose D.M., Masmoudi N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 1287–1315 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beale J.T., Hou T.Y., Lowengrub J.S.: Growth rates for the linearized motion of fluid interfaces away from equilibrium. Commun. Pure Appl. Math. 46(9), 1269–1301 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen G.Q., Wang Y.G.: Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics. Arch. Rational Mech. Anal. 187(3), 369–408 (2008)

    Article  ADS  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. 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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Díaz J.I., Lerena M.B.: On the inviscid and non-resistive limit for the equations of incompressible magnetohydrodynamics. Math. Model. Methods Appl. Sci. 12(10), 1401–1419 (2002)

    Article  MATH  Google Scholar 

  7. Duvaut G., Lions J.L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Rational Mech. Anal. 46, 241–279 (1972)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Friedman A., Liu Y.: A free boundary problem arising in magnetohydrodynamic system. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(3), 375–448 (1995)

    MATH  MathSciNet  Google Scholar 

  10. Goedbloed, J.P.H., Poedts, S.: Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press, Cambridge, (2004)

  11. He C., Xin Z.: Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations. J. Funct. Anal. 227(1), 113–152 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  14. Li X., Wang D.: Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows. J. Differ. Equ. 251(6), 1580–1615 (2011)

    Article  ADS  MATH  Google Scholar 

  15. Lindblad H.: Well-posedness for the linearized motion of an incompressible liquid with free surface boundary. Commun. Pure Appl. Math. 56(2), 153–197 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lindblad H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. (2) 162(1), 109–194 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lindblad H., Nordgren K.H.: A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary. J. Hyperbolic Differ. Equ. 6(2), 407–432 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Padula, M., Solonnikov, V.A.: On the free boundary problem of magnetohydrodynamics. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385(41), 135–186, 236 (2010). Translation in J. Math. Sci. (N. Y.) 178(3), 313–344 (2011)

  19. Schmidt P.G.: On a magnetohydrodynamic problem of Euler type. J. Differ. Equ. 74(2), 318–335 (1988)

    Article  ADS  MATH  Google Scholar 

  20. Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36(5), 635–664 (1983)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Trakhinin Y.: The existence of current-vortex sheets in ideal compressible magnetohydrodynamics. Arch. Rational Mech. Anal. 191(2), 245–310 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Trakhinin Y.: On the well-posedness of a linearized plasma–vacuum interface problem in ideal compressible MHD. J. Differ. Equ. 249(10), 2577–2599 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 1301, 39–72 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. 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)

    Article  MATH  Google Scholar 

  26. 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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, China

    Chengchun Hao

  2. BCMIIS, Beijing, 100048, China

    Chengchun Hao

  3. Department of Mathematics and Statistics, Georgetown University, Washington, DC, 20057, USA

    Tao Luo

Authors
  1. Chengchun Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chengchun Hao.

Additional information

Communicated by L. Saint-Raymond

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hao, C., Luo, T. A Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows. Arch Rational Mech Anal 212, 805–847 (2014). https://doi.org/10.1007/s00205-013-0718-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-013-0718-5

Keywords

Search

Navigation