Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension

Authors:
A. Fasano, M. Primicerio, S. D. Howison and J. R. Ockendon

Journal:
Quart. Appl. Math. **48** (1990), 153-168

MSC:
Primary 35R35; Secondary 35K05, 80A20

DOI:
https://doi.org/10.1090/qam/1040239

MathSciNet review:
MR1040239

Full-text PDF Free Access

References | Similar Articles | Additional Information

**[1]**D. A. Tarzia,*A Bibliography on Moving-free Boundary Problems for the Heat Diffusion Equation*, Istituto Matematico ``U. Dini", Firenze, 1988**[2]**C. M. Elliott and J. R. Ockendon,*Weak and Variational Methods for Moving Boundary Problems*, Res. Notes Math.**59**, Pitman, Boston-London, 1982 MR**650455****[3]**A. Fasano and M. Primicerio,*General free boundary problems for the heat equation*, J. Math. Anal. Appl., Part I**57**, 694-723; (1977) Part II**58**, 202-231; (1977). Part III**59**, 1-14 (1977) MR**0487016****[4]**A. Fasano, S. Kamin, and M. Primicerio,*Regularity of weak solutions of one dimensional two-phase Stefan problems*, Ann. Mat. Pura Appl.**115**, 341-348 (1977) MR**0477460****[5]**A. Fasano, M. Primicerio, and A. A. Lacey,*New results on some classical parabolic free-boundary problems*, Quart. Appl. Math.**38**, 439-460 (1981) MR**614552****[6]**B. Sherman,*A general one-phase Stefan problem*, Quart. Appl. Math.**28**, 377-382 (1970) MR**0282082****[7]**C. Baiocchi and G. A. Pozzi,*An evolution variational inequality related to a diffusion-absorption problem*, Appl. Math Optim.**2**, 304-314 (1975) MR**0430538****[8]**P. Van Moerbeke,*Optimal stopping and free boundary problems*, Rocky Mountain J. Math.**4**, 539-578 (1974) MR**0381188****[9]**A. Friedman,*Parabolic variational inequalities in one space dimension and smoothness of the free boundary*, J. Funct. Anal.**18**, 151-176 (1975) MR**0477461****[10]**E. Di Benedetto and A. Friedman,*The ill-posed Hele-shaw model and the Stefan problem for supercooled water*, Trans. Amer. Math. Soc.**282**, 183-204 (1984) MR**728709****[11]**R. Jensen,*Smoothness of the free boundary in the Stefan problem with supercooled water*, Illinois J. Math.**22**, 623-629 (1978) MR**503966****[12]**J. S. Langer,*Instabilities and pattern formation in crystal growth*, Rev. Modern Phys.**52**, 1-28 (1980)**[13]**A. Visintin,*supercooling and superheating effects in phase transition*, IMA J. Appl. Math.**35**, 233-256 (1985) MR**839201****[14]**A. Visintin,*A new model for supercooling and superheating effects*, IMA J. Appl. Math.**36**, 141-157 (1986) MR**984464****[15]**A. B. Crowley and J. R. Ockendon,*Modelling mushy regions*, Appl. Sci. Res.**44**, 1-7 (1987)**[16]**A. Schatz,*Free boundary problems of Stefan type with prescribed flux*, J. Math. Anal. Appl.**28**, 569-580 (1969) MR**0267285****[17]**A. Fasano,*Alcune osservazioni su una classe di problemi a contorno libero per l'equazione del calore*, Le Matematiche**29**, 397-411 (1974) MR**0410085****[18]**J. Crank and R. S. Gupta,*A moving boundary problem arising from the diffusion of oxygen in absorbing tissue*, J. Inst. Math. Appl.**10**, 19-33 (1972) MR**0347105****[19]**D. R. Atthey,*A finite difference scheme for melting problems*, J. Inst. Math. Appl.**13**, 353-356 (1974) MR**0351295****[20]**M. Primicerio,*Mushy region in phase change problems*, Applied Nonlinear Functional Analysis: Variational Methods and Ill-Posed Problems, K. H. Hoffmann and R. Gorenflo (Eds.), Verlag Peter Lang, Frankfurt, 1983, pp. 251-269 MR**685609****[21]**A. M. Meirmanov,*An example of nonexistence of a classical solution of the Stefan problem*, Soviet Math. Dokl.**23**, 564-566 (1981) MR**620870****[22]**S. D. Howison, A. A. Lacey, and J. R. Ockendon,*Singularity development in moving-boundary problems*, Quart. J. Mech. Appl. Math.**38**, 343-360 (1985) MR**800769****[23]**S. D. Howison,*Cusp development in Hele-Shaw flow with a free surface*, SIAM J. Appl. Math.**46**, 20-26 (1986) MR**821438****[24]**A. Fasano and M. Primicerio,*A critical case for the solvability of Stefan-like problems*, Math. Methods Appl. Sci.**5**, 84-96 (1983) MR**690897****[25]**A. B. Crowley,*Numerical solution of alloy solidification problems revisited*, Free Boundary Problems: Applications and Theory (A. Bossavit et al. eds.), Res. Notes Math.**120**, Pitman, 1985, pp. 122-131**[26]**W. Xie,*The Stefan problem with a kinetic condition at the free boundary*, to appear MR**1038897****[27]**D. G. Schaeffer,*Some examples of singularities in a free boundary*, Ann. Sc. Norm. Sup. Pisa (IV)**4**, 133-144 (1977) MR**0516201**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35R35,
35K05,
80A20

Retrieve articles in all journals with MSC: 35R35, 35K05, 80A20

Additional Information

DOI:
https://doi.org/10.1090/qam/1040239

Article copyright:
© Copyright 1990
American Mathematical Society