Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-03T02:42:51.624Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of some solutions of the Stefan problem†

Published online by Cambridge University Press:  16 July 2009

Riccardo Ricci  and
Xie Weiqing
Riccardo Ricci
Affiliation:
Dipartimento di Matematica, Universita di Milano, Via C. Saldini, 50-20133 Milano, Italy
Xie Weiqing
Affiliation:
Department of Mathematics, University of Pittsburgh, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the stability of travelling wave solutions of the one-dimensional under-cooled Stefan problem. We find a necessary and sufficient condition on the initial datum under which the free boundary is asymptotic to a travelling wave front. The method applies also to other types of solutions.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 1 , March 1991 , pp. 1 - 15
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bandle, C. & Stakgold, I. 1984 The formation of the dead core in parabolic reaction—diffusion equations. Trans. Amer. Math. Soc. 286, 275–293.Google Scholar
Brauner, C. M., Noor, Ebad S. & Schmidt-Lainé, C. 1989 Sur la stabilité d'ondes singulières en combustion. C.R. Acad. Sci. Paris, Série 1308, 159–162.Google Scholar
Buckmaster, J. & Ludford, G. S. S. 1983 Lectures on Mathematical Combustion. C.B.M.S.—N.S.F. Conference Series in Applied Math. 43, SlAM.Google Scholar
Chadam, J. & Ortoleva, P. 1983 The stabilizing effects of surface tension on the development of the free boundary in a planar one-dimensional Cauchy—Stefan problem. IMA J. Appl. Math. 30, 57–66.CrossRefGoogle Scholar
Dewynne, J. N., Howison, S. D., Ockendon, J. R. & Xie, W. 1989 Asymptotic behaviour of solutions to the Stefan problem with a kinetic condition at the free boundary. J. Austral. Math. Soc., Ser. B 31, 81–96.Google Scholar
Elliott, C. M. & Ockendon, J. R. 1982 Weak and Variational Methods for Moving Boundary Problems. Research Notes in Math. Vol. 59, Pitman.Google Scholar
Fasano, A., Mayer, G. & Primicerio, M. 1986 On a problem in polymer industry: Theoretical and numerical investigation. SIAM J. Math. Anal. 17, 945–960.CrossRefGoogle Scholar
Fasano, A. & Primicerio, M. 1983 A critical case for the solvability of Stefan-like problems. Math. Meth. in Appl. Sci. 5, 84–96.Google Scholar
Fasano, A., Primicerio, M., Howison, S. D. & Ockendon, J. R. 1990 On the singularities of one- dimensional Stefan problem with supercooling. In: Rodrigues, J. F. ed., Mathematical Models for Phase Change Problems, Birkhäuser.Google Scholar
Fasano, A. & Ricci, R. 1985 Penetration of solvents into glassy polymers. In: Bossavit, A., Damlamian, A. and Fremond, M., Free Boundary Problems: Applications and Theory, Research Notes in Math. Vol. 120, Pitman.Google Scholar
Hilhorst, D. & Hulshof, J. 1990 An elliptic—parabolic problem in combustion theory: convergence to travelling waves. Preprint. Université de Paris—Sud Mathématiques, 90–03.Google Scholar
Landau, H. G. 1950 Heat conduction in a melting solid. Quarterly Appl. Math. VIII, 81–94.Google Scholar
Mihačlov, V. P. 1970 On the stabilization of the solution of the Cauchy problem for the heat conduction equation. Soviet Math. Dokl. 11, 34–37.Google Scholar
Ricci, R. 1989 Travelling wave solutions of the Stefan and the ablation problems. SIAM J. Math. Anal. (to appear).Google Scholar
Sattinger, D. 1977 Weighted norms for the stability of traveling waves. J. Diff. Eqs 25, 130–144.Google Scholar
Umantsev, A. R. 1985 Motion of a plane front during cristallization. Kristallografiya 30, 153–160.Google Scholar