Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

The birth of a cusp in the two-dimensional, undercooled Stefan problem


Authors: M. A. Herrero, E. Medina and J. J. L. Velázquez
Journal: Quart. Appl. Math. 58 (2000), 473-494
MSC: Primary 35R35; Secondary 35B40, 35C20, 35K05
DOI: https://doi.org/10.1090/qam/1770650
MathSciNet review: MR1770650
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the one-phase, undercooled Stefan problem, in space dimension $ N = 2$. We show herein that planar, one-dimensional blow-up behaviours corresponding to the undercooling parameter $ \Delta = 1$ are unstable with respect to small, transversal perturbations. The solutions thus produced are shown to generically generate cusps in finite time, when they exhibit an undercooling $ \Delta = 1 - O\left( \epsilon \right) < 1$, where $ 0 < \epsilon << 1$, and $ \epsilon $ is a parameter that measures the strength of the perturbation. The asymptotic behaviour of solutions and interfaces near their cusps is also obtained. All results are derived by means of matched asymptotic expansions techniques.


References [Enhancements On Off] (What's this?)

  • [1] D. Andreucci, M. A. Herrero, and J. J. L. Velázquez, The classical one-phase Stefan problem: A catalogue of interface behaviours, to appear
  • [2] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139, 155-184 (1977) MR 0454350
  • [3] B. Caroli, C. Caroli, and B. Roulet, Instabilities of planar solidification fronts, In Solids Far from Equilibrium, C. Godrèche, editor, Cambridge University Press, 1992, pp. 155-296
  • [4] A. Fasano, M. Primicerio, S. W. Howinson, and J. R. Ockendon, On the singularities of one-dimensional Stefan problems with undercooling, In Mathematical Models for Phase Change Problems (J. F. Rodrigues, ed.), International Series on Numerical Mathematics, vol. 88, Birkhäuser, 1989, pp. 215-226.
  • [5] M. A. Herrero and J. J. L. Velázquez, Singularity formation in the one-dimensional supercooled Stefan problem, European Journal of Applied Math. 7, 119-150 (1996) MR 1388108
  • [6] A. A. Lacey, Bounds on the solutions of one-phase Stefan problems, European Journal of Applied Math. 6, 509-516 (1995) MR 1363760
  • [7] A. A. Lacey and J. R. Ockendon, Ill-posed free boundary problems, Control and Cybernetics 14, 275-296 (1985) MR 839524
  • [8] W. W. Mullins and R. F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, Journal of Applied Physics 35, 444-451 (1964)
  • [9] P. Pelcé (editor), Dynamics of Curved Fronts, Academic Press, 1988 MR 986791
  • [10] B. Sherman, A general one-phase Stefan problem, Quart. Appl. Math. 28, 377-383 (1970) MR 0282082
  • [11] J. J. L. Velázquez, Cusp formation for the undercooled Stefan problem in two and three dimensions, European Journal of Applied Math. 8, 1-21 (1997) MR 1431410

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35R35, 35B40, 35C20, 35K05

Retrieve articles in all journals with MSC: 35R35, 35B40, 35C20, 35K05


Additional Information

DOI: https://doi.org/10.1090/qam/1770650
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society