Asymptotic convergence of the Stefan problem to Hele-Shaw

Authors:
Fernando Quirós and Juan Luis Vázquez

Journal:
Trans. Amer. Math. Soc. **353** (2001), 609-634

MSC (2000):
Primary 35B40, 35R35

DOI:
https://doi.org/10.1090/S0002-9947-00-02739-2

Published electronically:
October 23, 2000

MathSciNet review:
1804510

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 , and give the precise asymptotic growth rate for the radius.

**[C]**L.A. Caffarelli,*The regularity of free boundaries in higher dimensions*, Acta Math.**139**(1977), 155-184. MR**56:12601****[CF]**L.A. Caffarelli and A. Friedman,*Continuity of the temperature in the Stefan problem*, Indiana Univ. Math. J.**28**(1) (1979), 53-70. MR**80i:35104****[D]**G. Duvaut,*Résolution d'un problème de Stefan (Fusion d'un bloc de glace à zéro degrés)*, C. R. Acad. Sci. Paris Sér. A**276**(1973), 1461-1463. MR**48:6688****[EJ]**C.M. Elliott and V. Janovský,*A variational inequality approach to Hele-Shaw flow with a moving boundary*, Proc. Royal Soc. Edinburgh**88A**(1981), 93-107. MR**82d:76031****[F]**A. Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Englewood Cliffs, NJ, 1964. MR**31:6062****[FK]**A. Friedman and D. Kinderlehrer,*A one phase Stefan problem*, Indiana Univ. Math. Jour.**24**(11) (1975), 1005-1035. MR**52:6190****[KLV]**J.R. King, A.A. Lacey and J.L. Vázquez,*Persistence of corners in free boundaries in Hele-Shaw flow*, European Jnl. Appl. Math.**6**(1995), 445-490. MR**97a:76037****[KN]**D. Kinderlehrer and L. Nirenberg,*The smoothness of the free boundary in the one phase Stefan problem*, Comm. Pure Appl. Math.**31**(1978), 257-282. MR**82b:35152****[L]**A.A. Lacey,*Bounds on solutions of one-phase Stefan problems*, European Jour. Appl. Math.**6**(1995), 509-516. MR**96j:80009****[LR]**B. Louro and J.F. Rodrigues,*Remarks on the quasi-steady one phase Stefan problem*, Proc. Royal Soc. Edinburgh**102A**(1986), 263-275. MR**88e:35186****[M]**H. Matano,*Asymptotic behavior of the free boundaries arising in one phase Stefan problems in multi-dimensional spaces*, in ``Nonlinear Partial Differential Equations in Applied Science'' (Tokyo, 1982), North-Holland Math. Stud., 81, 1983, pp. 133-151. MR**86a:35151****[Me]**A.M. Meirmanov,*The Stefan problem*, Walter de Gruyter, Berlin, 1992. MR**92m:35282****[MGR]**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.**[R]**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.**[Ru]**L.I. Rubinstein,*The Stefan problem*, Transl. Math. Monographs, vol. 27, Amer. Math. Soc., Providence, RI, 1971. MR**50:3837****[ST]**P.G. Saffman and G.I. Taylor,*The penetration of fluid into a porous medium Hele-Shaw cell containing a more viscous liquid*, Proc. Royal Soc. London Ser.**A 245**(1958), 312-329. MR**20:3697****[V]**J.L. Vázquez,*Singular solutions and asymptotic behaviour of nonlinear parabolic equations*, in ``International Conference on Differential Equations; Barcelona 91" (Equadiff-91), (C. Perelló, C. Simó and J. Solà-Morales eds.), World Scientific, Singapore, 1993, pp. 234-249. MR**95a:35067**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35B40,
35R35

Retrieve articles in all journals with MSC (2000): 35B40, 35R35

Additional Information

**Fernando Quirós**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain

**Juan Luis Vázquez**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain

DOI:
https://doi.org/10.1090/S0002-9947-00-02739-2

Keywords:
Stefan problem,
Hele-Shaw,
asymptotic behaviour

Received by editor(s):
December 31, 1996

Received by editor(s) in revised form:
October 28, 1997

Published electronically:
October 23, 2000

Article copyright:
© Copyright 2000
American Mathematical Society