Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The ill-posed Hele-Shaw model and the Stefan problem for supercooled water


Authors: Emmanuele DiBenedetto and Avner Friedman
Journal: Trans. Amer. Math. Soc. 282 (1984), 183-204
MSC: Primary 35R35; Secondary 35K05, 80A20
DOI: https://doi.org/10.1090/S0002-9947-1984-0728709-6
MathSciNet review: 728709
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Hele-Shaw flow of a slow viscous fluid between slightly separated plates is analyzed in the ill-posed case when the fluid recedes due to absorption through a core $ G$. 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 one-phase Stefan problem for supercooled water.


References [Enhancements On Off] (What's this?)

  • [1] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155-184. MR 0454350 (56:12601)
  • [2] -, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427-448. 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), 1297-1299. MR 607212 (82i:35078)
  • [4] L. A. Caffarelli and N. M. Riviere, Smoothness and analyticity of free boundaries in variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), 289-310. MR 0412940 (54:1061)
  • [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), 45-48. MR 0333443 (48:11768)
  • [6] P. Čižek and V. Janovsky, Hele-Shaw flow model of the injection by a point source, Proc. Roy. Sci. Edinburgh Sect. A 91 (1981), 147-159. MR 648924 (83h:76029)
  • [7] A. B. Crowley, On the weak solution of moving boundary problems, J. Inst. Math. Appl. 24 (1979), 43-57. 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), 121-131. MR 571974 (81f:82010)
  • [9] C. M. Elliott and V. Janovsky, A variational inequality approach to the Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 93-107. MR 611303 (82d:76031)
  • [10] C. M. Elliott and J. R. Ockendon, Weak and variational methods for moving boundary problems, Pittman, London, 1982. MR 650455 (83i:35157)
  • [11] A. Fasano and M. Primicerio, New results on some classical parabolic free boundary problems, Quart. Appl. Math. 38 (1980/81), 439-460. 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), 134-160. MR 0353110 (50:5596)
  • [14] -, Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal. 61 (1976), 97-125. 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), 1-24. MR 473315 (82i:35100)
  • [17] A. Friedman and D. Kinderlehrer, A one-phase Stefan problem, Indiana Univ. Math. J. 24 (1975), 1005-1035. 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), 623-629. MR 503966 (81g:35057)
  • [19] J. A. McGeough and H. Rasmussen, On the derivation of the quasi-steady model in electrochemical machining, J. Inst. Math. Appl. 13 (1974), 13-21.
  • [20] 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 (1981), 455-469.
  • [21] P. van Moerbecke, An optimal stopping problem for linear reward, Acta Math. 132 (1974), 1-41. MR 0376225 (51:12405)
  • [22] S. Richardson, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972), 609-618.
  • [23] -, Some Hele-Shaw flows with time-dependent free boundaries, J. Fluid Mech. 102 (1981), 263-278. MR 612095 (82d:76023)
  • [24] P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid, Proc. Roy. Soc. Edinburgh Sect. A 245 (1958), 312-329. 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), 266-269. MR 0390499 (52:11325)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35R35, 35K05, 80A20

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


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0728709-6
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society