Instability of periodic states for the Sivashinsky equation
Author:
A. Novick-Cohen
Journal:
Quart. Appl. Math. 48 (1990), 217-224
MSC:
Primary 35B10; Secondary 35K55, 35Q99, 76E99, 80A22
DOI:
https://doi.org/10.1090/qam/1052132
MathSciNet review:
MR1052132
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Abstract: The Sivashinsky equation is an asymptotically derived model equation for evolution of the solid-liquid interface which occurs during directional solidification of dilute binary alloys. During the solidification process interfaces are known experimentally to yield planar, cellular, cusped, or dendritic structures. Cellular structures, interpreted here as periodic one dimensional nontrivial steady states, are shown in this paper to be unstable, if they exist, within the context of the Sivashinsky equation. Symmetric nontrivial steady states are likewise shown to be unstable.
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G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy, Physica 8D, 243–248 (1983)
A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys, preprint
C. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96, 339–357 (1986)
G. W. Young and S. H. Davis. Systems with small segregation coefficient, Phys. Rev. B 34, 3388–3396 (1986)
N. Chafee, Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions, J. Differential Equations 18, 111–134 (1975)
J. Carr, M. E. Gurtin, and M. Slemrod, Structural phase transitions on a finite interval, Arch. Rational Mech. Anal. 86, 317–351 (1984)
M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977
G. W. Young, S. H. Davis, and K. E. Brattkus, Anisotropic interface kinetics and tilted cells in unidirectional solidification, J. Crystal Growth 83, 560–571 (1988)
K. Brattkus and S. H. Davis, Cellular growth near absolute stability, Phys. Rev. B38, 11,452–11,460 (1988)
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© Copyright 1990
American Mathematical Society
