Asymptotic convergence of the Stefan problem to Hele-Shaw

Quirós, Fernando; Vázquez, Juan

doi:10.1090/S0002-9947-00-02739-2

Asymptotic convergence of the Stefan problem to Hele-Shaw
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by Fernando Quirós and Juan Luis Vázquez PDF

Trans. Amer. Math. Soc. 353 (2001), 609-634 Request permission

Abstract:

We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as $t\to \infty$, and give the precise asymptotic growth rate for the radius.

References

Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), no. 3-4, 155–184. MR 454350, DOI 10.1007/BF02392236
Luis A. Caffarelli and Avner Friedman, Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J. 28 (1979), no. 1, 53–70. MR 523623, DOI 10.1512/iumj.1979.28.28004
Georges Duvaut, Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1461–A1463 (French). MR 328346
C. M. Elliott and V. Janovský, A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 1-2, 93–107. MR 611303, DOI 10.1017/S0308210500017315
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Avner Friedman and David Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J. 24 (1974/75), no. 11, 1005–1035. MR 385326, DOI 10.1512/iumj.1975.24.24086
J. R. King, A. A. Lacey, and J. L. Vázquez, Persistence of corners in free boundaries in Hele-Shaw flow, European J. Appl. Math. 6 (1995), no. 5, 455–490. Complex analysis and free boundary problems (St. Petersburg, 1994). MR 1363758, DOI 10.1017/S0956792500001984
David Kinderlehrer and Louis Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Comm. Pure Appl. Math. 31 (1978), no. 3, 257–282. MR 480348, DOI 10.1002/cpa.3160310302
A. A. Lacey, Bounds on solutions of one-phase Stefan problems, European J. Appl. Math. 6 (1995), no. 5, 509–516. Complex analysis and free boundary problems (St. Petersburg, 1994). MR 1363760, DOI 10.1017/S095679250000200X
Bento Louro and José-Francisco Rodrigues, Remarks on the quasisteady one phase Stefan problem, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 3-4, 263–275. MR 852360, DOI 10.1017/S0308210500026354
Hiroshi Matano, Asymptotic behavior of the free boundaries arising in one-phase Stefan problems in multidimensional spaces, Nonlinear partial differential equations in applied science (Tokyo, 1982) North-Holland Math. Stud., vol. 81, North-Holland, Amsterdam, 1983, pp. 133–151. MR 730240
Anvarbek M. Meirmanov, The Stefan problem, De Gruyter Expositions in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1992. Translated from the Russian by Marek Niezgódka and Anna Crowley; With an appendix by the author and I. G. Götz. MR 1154310, DOI 10.1515/9783110846720.245
J.A. McGeough and H. Rasmussen, On the derivation of the quasi-steady model in electromechanical machining, J. Inst. Math. Applics. 13 (1974), 13–21.
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.
L. I. Rubenšteĭn, The Stefan problem, Translations of Mathematical Monographs, Vol. 27, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by A. D. Solomon. MR 0351348
P. G. Saffman and Geoffrey Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London Ser. A 245 (1958), 312–329. (2 plates). MR 97227, DOI 10.1098/rspa.1958.0085
Juan Luis Vázquez, Singular solutions and asymptotic behaviour of nonlinear parabolic equations, International Conference on Differential Equations, Vol. 1, 2 (Barcelona, 1991) World Sci. Publ., River Edge, NJ, 1993, pp. 234–249. MR 1242245