[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content

Local Existence of 2D Compressible Current-Vortex Sheets

  • Conference paper
  • First Online:
Hyperbolic Problems: Theory, Numerics, Applications. Volume I (HYP 2022)

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 34))

  • 234 Accesses

Abstract

This survey presents our recent results in Morando et al. 2023, where we establish the local-in-time existence and the nonlinear stability of current-vortex sheets in ideal compressible Magnetohydrodynamics, provided a suitable stability condition is satisfied at each point of the initial discontinuity.

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

Access this chapter

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 119.50
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
GBP 149.99
Price includes VAT (United Kingdom)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Alinhac, S.: Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Commun. Partial Differ. Eqs. 14(2), 173–230 (1989)

    Article  Google Scholar 

  2. Alinhac, S., Gérard, P.: Pseudo-differential Operators and the Nash-Moser Theorem. American Mathematical Society, Providence (2007)

    Book  Google Scholar 

  3. Benzoni-Gavage, S., Serre, D.: Multidimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Chen, S.: Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary. Transl. Chin. Ann. Math. 3(2), 222–232 (1982), Front. Math. China 2(1), 87-102 (2017)

    Google Scholar 

  6. Coulombel, J.-F., Secchi, P.: The stability of compressible vortex sheets in two space dimensions. Indiana Univ. Math. J. 53, 941–1012 (2004)

    Article  MathSciNet  Google Scholar 

  7. Coulombel, J.-F., Secchi, P.: Nonlinear compressible vortex sheets in two space dimensions. Ann. Sci. Ec. Norm. Super. 41, 85–139 (2008)

    Article  MathSciNet  Google Scholar 

  8. Fejer, J.A., Miles, W.: On the stability of a plane vortex sheet with respect to three dimensional disturbances. J. Fluid Mech. 15, 335–336 (1963)

    Article  MathSciNet  Google Scholar 

  9. Michael, D.H.: The stability of a combined current and vortex sheet in a perfectly conducting fluid. Proc. Cambridge Philos. Soc. 51, 528–532 (1955)

    Article  MathSciNet  Google Scholar 

  10. Miles, J.W.: On the disturbed motion of a plane vortex sheet. J. Fluid Mech. 4, 538–552 (1958)

    Article  MathSciNet  Google Scholar 

  11. Morando, A., Secchi, P., Trebeschi, P.: Regularity of solutions to characteristic initial-boundary value problems for symmetrizable systems. J. Hyper. Differ. Equ. 6(4), 753–808 (2009)

    Article  MathSciNet  Google Scholar 

  12. Morando, A., Secchi, P., Trebeschi, P., Yuan, D.: Nonlinear stability and existence of two-dimensional compressible current-vortex sheets. Arch. Rational Mech. Anal. 247, 50 (2023). https://doi.org/10.1007/s00205-023-01865-w

  13. Ohno, M., Shizuta, Y., Yanagisawa, T.: The initial boundary value problem for linear symmetric hyperbolic problems with boundary characteristic of constant multiplicity. J. Math. Kyoto Univ. 35, 143–210 (1995)

    MathSciNet  Google Scholar 

  14. Rauch, J.: Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. Am. Math. Soc. 291, 167–187 (1985)

    Article  MathSciNet  Google Scholar 

  15. Secchi, P.: The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity. Differ. Integr. Equ. 9(4), 671–700 (1996)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Secchi, P.: On the Nash-Moser iteration technique. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds.) Recent Developments of Mathematical Fluid Mechanics, pp. 443–457. Birkhäuser, Basel (2016)

    Chapter  Google Scholar 

  18. Trakhinin, Y.: Existence of compressible current-vortex sheets: variable coefficients linear analysis. Arch. Ration. Mech. Anal. 177(3), 331–366 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Tsuji, M.: Regularity of solutions of hyperbolic mixed problems with characteristic boundary. Proc. Jpn. Acad. 48, 719–724 (1972)

    MathSciNet  Google Scholar 

  21. Wang, Y.-G., Yu, F.: Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets. Arch. Ration. Mech. Anal. 208, 341–389 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alessandro Morando .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Morando, A., Secchi, P., Trebeschi, P., Yuan, D. (2024). Local Existence of 2D Compressible Current-Vortex Sheets. In: Parés, C., Castro, M.J., Morales de Luna, T., Muñoz-Ruiz, M.L. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Volume I. HYP 2022. SEMA SIMAI Springer Series, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-031-55260-1_24

Download citation

Publish with us

Policies and ethics