The illposed HeleShaw model and the Stefan problem for supercooled water
Authors:
Emmanuele DiBenedetto and Avner Friedman
Journal:
Trans. Amer. Math. Soc. 282 (1984), 183204
MSC:
Primary 35R35; Secondary 35K05, 80A20
MathSciNet review:
728709
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Abstract: The HeleShaw flow of a slow viscous fluid between slightly separated plates is analyzed in the illposed case when the fluid recedes due to absorption through a core . Necessary and sufficient conditions are given on the initial domain occupied by the fluid to ensure the existence of a solution. Regularity of the free boundary is established in certain rather general cases. Similar results are obtained for the analogous parabolic version, which models the onephase Stefan problem for supercooled water.
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 [1]
 L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155184. MR 0454350 (56:12601)
 [2]
 , Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427448. MR 567780 (81e:35121)
 [3]
 , A remark on the Hausdorff measure of a free boundary and the convergence of coincidence sets, Boll. Un. Mat. Ital. 18 (1981), 12971299. MR 607212 (82i:35078)
 [4]
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 [5]
 J. R. Cannon, D. B. Henry and D. B. Kotlow, Continuous differentiability of the free boundary for weak solutions of the Stefan problem, Bull. Amer. Math. Soc. 80 (1974), 4548. MR 0333443 (48:11768)
 [6]
 P. Čižek and V. Janovsky, HeleShaw flow model of the injection by a point source, Proc. Roy. Sci. Edinburgh Sect. A 91 (1981), 147159. MR 648924 (83h:76029)
 [7]
 A. B. Crowley, On the weak solution of moving boundary problems, J. Inst. Math. Appl. 24 (1979), 4357. MR 539372 (80d:65129)
 [8]
 C. M. Elliott, On a variational inequality formulation of an electrical machining moving boundary problem and its approximation by the finite element method, J. Inst. Math. Appl. 25 (1980), 121131. MR 571974 (81f:82010)
 [9]
 C. M. Elliott and V. Janovsky, A variational inequality approach to the HeleShaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 93107. MR 611303 (82d:76031)
 [10]
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 [11]
 A. Fasano and M. Primicerio, New results on some classical parabolic free boundary problems, Quart. Appl. Math. 38 (1980/81), 439460. MR 614552 (82g:35061)
 [12]
 A. Friedman, Partial differential equations of parabolic type, Prentice Hall, Englewood Cliffs, N.J., 1964. MR 0181836 (31:6062)
 [13]
 , Parabolic variational inequalities in unbounded domains and applications to stopping time problems, Arch. Rational Mech. Anal. 52 (1973), 134160. MR 0353110 (50:5596)
 [14]
 , Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal. 61 (1976), 97125. MR 0407452 (53:11227)
 [15]
 , Variational principles and free boundary problems, Wiley, New York, 1982. MR 679313 (84e:35153)
 [16]
 A. Friedman and R. Jensen, Convexity of the free boundary in the Stefan problem and in the dam problem, Arch. Rational Mech. Anal. 67 (1977), 124. MR 473315 (82i:35100)
 [17]
 A. Friedman and D. Kinderlehrer, A onephase Stefan problem, Indiana Univ. Math. J. 24 (1975), 10051035. MR 0385326 (52:6190)
 [18]
 R. Jensen, The smoothness of the free boundary in the Stefan problem with supercooled water, Illinois J. Math. 22 (1978), 623629. MR 503966 (81g:35057)
 [19]
 J. A. McGeough and H. Rasmussen, On the derivation of the quasisteady model in electrochemical machining, J. Inst. Math. Appl. 13 (1974), 1321.
 [20]
 J. W. McLean and P. G. Saffman, The effect of surface tension on the shape of fingers in a HeleShaw cell, J. Fluid Mech. 102 (1981), 455469.
 [21]
 P. van Moerbecke, An optimal stopping problem for linear reward, Acta Math. 132 (1974), 141. MR 0376225 (51:12405)
 [22]
 S. Richardson, HeleShaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972), 609618.
 [23]
 , Some HeleShaw flows with timedependent free boundaries, J. Fluid Mech. 102 (1981), 263278. MR 612095 (82d:76023)
 [24]
 P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium or HeleShaw cell containing a more viscous fluid, Proc. Roy. Soc. Edinburgh Sect. A 245 (1958), 312329. MR 0097227 (20:3697)
 [25]
 D. G. Schaeffer, A new proof of the infinite differentiability of the free boundary in the Stefan problem, J. Differential Equations 20 (1976), 266269. MR 0390499 (52:11325)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407287096
PII:
S 00029947(1984)07287096
Article copyright:
© Copyright 1984
American Mathematical Society
