Some criteria for the disappearance of the mushy region in the Stefan problem
Authors:
I. G. Götz and B. Zaltzman
Journal:
Quart. Appl. Math. 53 (1995), 657-671
MSC:
Primary 35R35; Secondary 35K05, 80A22
DOI:
https://doi.org/10.1090/qam/1359501
MathSciNet review:
MR1359501
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Abstract: The disappearance of the mushy region in a multidimensional one-phase Stefan problem is discussed. In the case of a piecewise-smooth boundary of the domain and bounded initial-boundary data, sufficient conditions for the disappearance of the mushy zone in a finite time are presented. For a ${C^2}$ -smooth boundary and appropriately smooth boundary data both necessary and sufficient conditions for the mush to vanish are obtained. Possible behaviors of the transient phase for a twodimensional solution near a corner point of the domain are also investigated.
D. R. Atthey, A finite difference scheme for melting problems, J. Inst. Math. Appl. 354 (1975)
J. I. Diaz, A. Fasano, and A. Meirmanov, On the disappearance of the mushy region in multidimensional Stefan problem, Proc. Free Boundary Problems: Theory & Applications, Montreal, 1990
I. G. Götz and B. Zaltzman, On the behavior of mushy region in a Stefan problem, Internat. Ser. Numer. Math. 9, 155–163 (1991)
I. G. Götz and B. Zaltzman, Nonincrease of mushy region in a nonhomogeneous Stefan problem, Quart. Appl. Math. XLIX, 741–746 (1991)
B. Gustafsson and J. Mossino, Quelques inequalités isopérimétriques pour le problème de Stefan, Math. Analysis, C. R. Acad. Sci. Paris 305 (1), 669–672 (1987)
S. L. Kamenomostskaya, On the Stefan problem, Mat. Sbornik. 53 (95), 489–514 (1961) (in Russian)
O. A. Ladyženskaja and N. N. Ural’ceva, Linear and quasilinear equations of elliptic type, Nauka: Moscow (1967) (in Russian)
A. M. Meirmanov, An example of nonexistence of a classical solution of the Stefan problem, Soviet Math. Dokl. 3 (28), 564–566 (1981)
A. M. Meirmanov, The structure of a generalized solution of the Stefan problem, Periodic solutions, Dokl. Akad. Nauk SSSR 272 (4), 789–791 (1981); English transl. in Soviet Math. Dokl. 2 (28) 440–443 (1983)
A. M. Meirmanov, On the disappearance of the mushy region in the Stefan problem with spherical symmetry (in Russian), Dinamika sploshnoi sredy, Lavrent’ev Institute of Hydrodynamics 91, 86–99 (1985)
A. M. Meirmanov, The Stefan problem, De Gruyter Expositions in Math., Berlin, 244 (1992)
O. A. Oleinik, A method of solution of the general Stefan problem, Soviet Math. Dokl. 1, 1350–1354 (1960)
M. Primicerio, Mushy region in phase-change problem, Appl. Nonlinear Funct. Anal., Lang, Frankfurt/Main, 251–269 (1982)
J. C. W. Rogers and A. E. Berger, Some properties of the nonlinear semigroup for the problem ${u_t} - Df\left ( u \right ) = 0$, Nonlinear Anal., Theory, Methods, and Applications 8 (8), 909–939 (1984)
D. R. Atthey, A finite difference scheme for melting problems, J. Inst. Math. Appl. 354 (1975)
J. I. Diaz, A. Fasano, and A. Meirmanov, On the disappearance of the mushy region in multidimensional Stefan problem, Proc. Free Boundary Problems: Theory & Applications, Montreal, 1990
I. G. Götz and B. Zaltzman, On the behavior of mushy region in a Stefan problem, Internat. Ser. Numer. Math. 9, 155–163 (1991)
I. G. Götz and B. Zaltzman, Nonincrease of mushy region in a nonhomogeneous Stefan problem, Quart. Appl. Math. XLIX, 741–746 (1991)
B. Gustafsson and J. Mossino, Quelques inequalités isopérimétriques pour le problème de Stefan, Math. Analysis, C. R. Acad. Sci. Paris 305 (1), 669–672 (1987)
S. L. Kamenomostskaya, On the Stefan problem, Mat. Sbornik. 53 (95), 489–514 (1961) (in Russian)
O. A. Ladyženskaja and N. N. Ural’ceva, Linear and quasilinear equations of elliptic type, Nauka: Moscow (1967) (in Russian)
A. M. Meirmanov, An example of nonexistence of a classical solution of the Stefan problem, Soviet Math. Dokl. 3 (28), 564–566 (1981)
A. M. Meirmanov, The structure of a generalized solution of the Stefan problem, Periodic solutions, Dokl. Akad. Nauk SSSR 272 (4), 789–791 (1981); English transl. in Soviet Math. Dokl. 2 (28) 440–443 (1983)
A. M. Meirmanov, On the disappearance of the mushy region in the Stefan problem with spherical symmetry (in Russian), Dinamika sploshnoi sredy, Lavrent’ev Institute of Hydrodynamics 91, 86–99 (1985)
A. M. Meirmanov, The Stefan problem, De Gruyter Expositions in Math., Berlin, 244 (1992)
O. A. Oleinik, A method of solution of the general Stefan problem, Soviet Math. Dokl. 1, 1350–1354 (1960)
M. Primicerio, Mushy region in phase-change problem, Appl. Nonlinear Funct. Anal., Lang, Frankfurt/Main, 251–269 (1982)
J. C. W. Rogers and A. E. Berger, Some properties of the nonlinear semigroup for the problem ${u_t} - Df\left ( u \right ) = 0$, Nonlinear Anal., Theory, Methods, and Applications 8 (8), 909–939 (1984)
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© Copyright 1995
American Mathematical Society
