Skip to main content
SpringerLink
Log in

Solvability of the localized induction equation for vortex motion

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The initial and the initial-boundary value problems for the localized induction equation which describes the motion of a vortex filament are considered. We prove the existence of solutions of both problems globally in time in the sense of distribution by the method of regularization.

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. Arms, R.J., Hama, F.R.: Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids8, 553–559 (1965)

    Google Scholar 

  2. Fukumoto, Y., Miyazaki, T.: N-solitons on a curved vortex filament. J. Phys. Soc. Japan55, 4152–4155 (1986)

    Google Scholar 

  3. Ginibre, J., Velo, G.: On a class of non-linear Schrödinger equations. III Special theories in dimensions 1, 2 and 3. Ann. Inst. H. Poincaré Sect. A28, 287–316 (1978)

    Google Scholar 

  4. Hama, F.R.: Progressive deformation of a curved vortex filament by its own induction. Phys. Fluids5, 1156–1162 (1962)

    Google Scholar 

  5. Hasimoto, H.: A soliton on a vortex filament. J. Fluid Mech.51, 477–485 (1972)

    Google Scholar 

  6. Hayashi, N., Nakamitsu, K., Tsutsumi, M.: On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimensions. Math. Z.192, 637–650 (1986)

    Google Scholar 

  7. Kida, S.: A vortex filament moving without change of form. J. Fluid Mech.112, 397–409 (1981)

    Google Scholar 

  8. Mizohata, S.: Theory of partial differential equations. Cambridge: Cambridge Univ. Press, 1973

    Google Scholar 

  9. Slobodetskiî, L.N.: Estimates of solutions of elliptic and parabolic systems. Dokl. Akad. Nauk SSSR120, 468–471 (1958) (in Russian)

    Google Scholar 

  10. Solonnikov, V.A.: On an initial-boundary value problem for the Stokes systems arising in the study of a problem with a free boundary. Proc. Steklov Inst. Math.188, 191–239 (1991)

    Google Scholar 

  11. Takaki, R.: Numerical analysis of distortion of a vortex filament. J. Phys. Soc. Japan38, 1530–1537 (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Keio University, 223, Yokohama, Japan

    Takahiro Nishiyama & Atusi Tani

Authors
  1. Takahiro Nishiyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Atusi Tani
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. Araki

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nishiyama, T., Tani, A. Solvability of the localized induction equation for vortex motion. Commun.Math. Phys. 162, 433–445 (1994). https://doi.org/10.1007/BF02101741

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02101741

Keywords

Search

Navigation