Skip to main content
SpringerLink
Log in

Convergence of Travelling Waves for Phase Field Equations to Sharp Interface Models in the Singular Limit

  • Conference paper
Patterns and Dynamics in Reactive Media

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 37))

  • 263 Accesses

Abstract

We show the convergence of travelling waves for the phase field equations to that of a modified Stefan model in an appropriate singular limit. Finite surface tension effect is crucial to prove this convergence.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Physical Review A 39 (1989), 5887–5896.

    Article  Google Scholar 

  2. G. Caginalp and J. Chadam, Stability of interfaces with velocity correction term, to appear in Rocky Mountain J. of Math.

    Google Scholar 

  3. G. Caginalp and P.C. Fife, Elliptic problems involving phase boundaries satisfying a curvature condition, IMA J. of Applied Math. 38 (1987), 195–217.

    Article  Google Scholar 

  4. G. Caginalp and P.C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math 48 (1988), 506–518.

    Google Scholar 

  5. G. Caginalp and Y. Nishiura, The existence of travelling waves for phase field equations and convergence to sharp interface models in the singular limit, To appear in Quarterly of Appl. Math.

    Google Scholar 

  6. S-N. Chow and J. Hale, Methods of Bifurcation Theory, Springer, Berlin (1982).

    Google Scholar 

  7. J.N. Dewynne, S.D. Howison, J.R. Ockendon, W. Xie, Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary, J. Austral. Math. Soc. Ser. B 31 (1989), 81–96.

    Article  Google Scholar 

  8. H. Fujii, Y. Nishiura, M. Mimura, R. Kobayashi, Existence of curved fronts for the phase field model, In preparation.

    Google Scholar 

  9. J.W. Wilder, Travelling wave solutions for interfaces arising from phase boundaries based on a phase field model, Rensselaer Polytechnics Inst. Preprint.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Hiroshima University, Hiroshima, 730, Japan

    Yasumasa Nishiura

  2. Mathematics Department, University of Pittsburgh, PA, 15620, USA

    Gunduz Caginalp

Authors
  1. Yasumasa Nishiura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gunduz Caginalp
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Chemical Engineering and Material Science, University of Minnesota, Minneapolis, MN, 55455, USA

    Rutherford Aris

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Donald G. Aronson

  3. Physics Department and the Center for Nonlinear Dynamics, University of Texas, Austin, TX, 78712, USA

    Harry L. Swinney

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag New York Inc.

About this paper

Cite this paper

Nishiura, Y., Caginalp, G. (1991). Convergence of Travelling Waves for Phase Field Equations to Sharp Interface Models in the Singular Limit. In: Aris, R., Aronson, D.G., Swinney, H.L. (eds) Patterns and Dynamics in Reactive Media. The IMA Volumes in Mathematics and its Applications, vol 37. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3206-3_9

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-3206-3_9

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7832-0

  • Online ISBN: 978-1-4612-3206-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation