Some remarks on the Stefan problem with surface structure
Authors:
Morton E. Gurtin and H. Mete Soner
Journal:
Quart. Appl. Math. 50 (1992), 291-303
MSC:
Primary 35R35; Secondary 35K05, 76A99, 76D45, 80A22
DOI:
https://doi.org/10.1090/qam/1162277
MathSciNet review:
MR1162277
Full-text PDF Free Access
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Abstract: This paper discusses a generalized Stefan problem which allows for supercooling and superheating and for capillarity in the interface between phases. Simple solutions are obtained indicating the chief differences between this problem and the classical Stefan problem. A weak formulation of the general problem is given.
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W. W. Mullins and R. F. Sekerka, Morphological stability of a particle growing by diffusion or heat flow, J. Appl. Phys. 34, 323–329 (1963)
J. W. McLean and P. G. Saffman, The effect of surface tension on the shape of fingers in a Hele-Shaw cell, J. Fluid. Mech. 102, 455–469 (1981)
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M. E. Gurtin , G. Kossioris, and H. M. Soner, The one-dimensional superthermal Stefan problem, forthcoming
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E. Yokoyama and T. Kuroda, Pattern formation in growth of snow crystals occurring in the surface kinetic process and the diffusion process, Phys. Rev. A(3) 41, 2038–2049 (1990)
W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27, 900–904 (1958)
P. G. Saffman and G. I. Taylor, The penetration of a fluid into porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London Ser. A 245, 312–329 (1958)
W. W. Mullins, Grain boundary grooving by volume diffusion, Trans. Met. Soc. AIME 218, 354–361 (1960)
W. W. Mullins and R. F. Sekerka, Morphological stability of a particle growing by diffusion or heat flow, J. Appl. Phys. 34, 323–329 (1963)
J. W. McLean and P. G. Saffman, The effect of surface tension on the shape of fingers in a Hele-Shaw cell, J. Fluid. Mech. 102, 455–469 (1981)
M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277, 1–42 (1983)
J. Duchon and R. Robert, Evolution d’une interface par capillarité et diffusion de volume, existence local en temps, C. R. Acad. Sci. Paris Sér. I Math. 298, 473–476 (1984)
G. Barles, Remark on a flame propagation model, Rapport INRIA, No. 464, 1985
J. A. Sethian, Curvature and the evolution of fronts, Comm. Math. Phys. 101, 487–499 (1985)
M. E. Gurtin, Multiphase thermomechanics with interfacial structure. 1. Heat conduction and the capillary balance law, Arch. Rational Mech. Anal. 104, 195–221 (1988)
S. Osher and J. A. Sethian, Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79, 12–49 (1988)
S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interface structure. 2. Evolution of an isothermal interface, Arch. Rational Mech. Anal. 108, 323–391 (1989)
Y. -G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geometry, to appear
L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Diff. Geometry, II, Trans. Amer. Math. Soc., to appear
M. E. Gurtin , G. Kossioris, and H. M. Soner, The one-dimensional superthermal Stefan problem, forthcoming
S. Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, European J. Appl. Math. 1, 101–111 (1990)
H. M. Soner, Motion of a set by the curvature of its boundary, J. Differential Equations, to appear
E. Yokoyama and T. Kuroda, Pattern formation in growth of snow crystals occurring in the surface kinetic process and the diffusion process, Phys. Rev. A(3) 41, 2038–2049 (1990)
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Article copyright:
© Copyright 1992
American Mathematical Society