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 Hele-Shaw problem as a ``Mesa'' limit of Stefan problems: Existence, uniqueness, and regularity of the free boundary

Authors: Ivan A. Blank, Marianne K. Korten and Charles N. Moore
Journal: Trans. Amer. Math. Soc. 361 (2009), 1241-1268
MSC (2000): Primary 76D27, 35K65, 49J40
Published electronically: October 10, 2008
MathSciNet review: 2457397
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study a Hele-Shaw problem with a mushy region obtained as a mesa type limit of one-phase Stefan problems in exterior domains. We deal with both Neumann and Dirichlet data and show pointwise convergence of the Stefan solutions to the Hele-Shaw solution. We make no assumptions on the geometry, topology, or connectivity of the injection slot.

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

  • [AK] D. Andreucci and M. Korten, Initial traces of solutions to a one-phase Stefan problem in an infinite strip, Rev. Mat. Iberoamericana, 9(1993), no. 2, 315-332. MR 1232846 (94m:35319)
  • [BF] B. Bazaliy and A. Friedman, Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: An application to a model of tumor growth, Indiana Univ. Math. J., 52(2003), no. 5, 1265-1304. MR 2010327 (2004j:35302)
  • [BBH] Ph. Bénilan, L. Boccardo, and M.A. Herrero, On the limit of solutions of $ u_t = \Delta u^m$ as $ m \rightarrow \infty,$ Rend. Mat. Univ. Pol. Torino, Fascicolo Speciale (1989), Nonl. PDE's, 1-13. MR 1155452 (93c:35069)
  • [BEG] Ph. Bénilan, L.C. Evans, and R.F. Gariepy, On some singular limits of homogeneous semigroups, J. Evol. Eq., 3(2003), no. 2, 203-214. MR 1980973 (2004d:34122)
  • [Bl] I. Blank, Sharp results for the regularity and stability of the free boundary in the obstacle problem, Indiana Univ. Math. J. 50(2001), no. 3, 1077-1112. MR 1871348 (2002i:35213)
  • [Bo] J.E. Bouillet, Nonuniqueness in $ L^{\infty}$: An example, Lecture notes in pure and applied mathematics, Vol. 148, Marcel Dekker, Inc., (1993). MR 1236685 (94h:35099)
  • [BoKM] J.E. Bouillet, M.K. Korten, and V. Márquez, Singular limits and the ``Mesa'' problem, Rev. Un. Mat. Argentina, 41(1998), no. 1, 27-40. MR 1682193 (2000g:35236)
  • [C] L.A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4(1998), no. 4-5, 383-402. MR 1658612 (2000b:49004)
  • [CF] L.A. Caffarelli and A. Friedman, Asymptotic behavior of solutions of $ u_t = \Delta u^{m}$ as $ m \rightarrow \infty,$ Indiana Univ. Math. J., 36(1987), no. 4, 203-224. MR 916741 (88m:35075)
  • [CJK] S. Choi, D. Jerison, and I. Kim, Regularity for the one-phase Hele-Shaw problem from a Lipschitz initial surface, Amer. J. Math. 129 (2007), 527-582. MR 2306045 (2008d:35238)
  • [DK] D. Danielli and M.K. Korten, On the pointwise jump condition at the free boundary in the 1-phase Stefan problem, Comm. Pure Appl. Anal., 4(2005), no. 2, 357-366. MR 2149521 (2006c:35304)
  • [DL] P. Daskalopoulos and K. Lee, All time smooth solutions of the one-phase Stefan problem and the Hele-Shaw flow, Comm. PDE, 29(2004), no. 1-2, 71-88. MR 2038144 (2004m:35283)
  • [DB] E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl., (4), CXXX(1982), 131-176. MR 663969 (83k:35045)
  • [DF] E. DiBenedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282(1984), no. 1, 183-204. MR 728709 (85g:35121)
  • [EHKO] C.M. Elliott, M.A. Herrero, J.R. King, and J.R. Ockendon, The mesa problem: Diffusion patterns for $ u_t = \nabla \cdot (u^m \nabla u)$ as $ m \rightarrow +\infty,$ IMA J. Appl. Math., 37(1986), no. 2, 147-154. MR 983523 (89m:76061)
  • [EJ] C.M. Elliott and V. Janovský, A variational inequality approach to the Hele-Shaw flow with a moving boundary, Proc. Royal Soc. Edinburgh, Sect. 88A, 93-107. MR 611303 (82d:76031)
  • [ES] J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28(1997), 1028-1047. MR 1466667 (98i:35213)
  • [FH] A. Friedman and S. Huang, Asymptotic behavior of solutions of $ u_t = \Delta \phi_m(u)$ as $ m \rightarrow \infty$ with inconsistent initial values, Analyse mathématique et applications, Gauthier-Villars, Paris, (1988), 165-180. MR 956958 (89m:35027)
  • [GQV] O. Gil, F. Quirós, and J.L. Vázquez, Zero specific heat limit and large time asymptotics for the one-phase Stefan problem, preprint.
  • [GH] D. Gilbarg and L. Hörmander, Intermediate Schauder estimates, Arch. Ration. Mech. Anal., 74(1980), no. 4, 297-318. MR 588031 (82a:35038)
  • [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 Edition, Springer-Verlag, New York, 2001. MR 1814364 (2001k:35004)
  • [I] N. Igbida, The mesa-limit of the porous-medium equation and the Hele-Shaw problem, Diff. Int. Eqns., 15(2002), no. 2, 129-146. MR 1870466 (2002m:35119)
  • [JK] D. Jerison and I. Kim, The one-phase Hele-Shaw problem with singularities, J. Geom. Anal., 15(2005), no. 4, 641-668. MR 2203166 (2006j:35247)
  • [Ki] I. Kim, Regularity of the free boundary for the one-phase Hele-Shaw problem, J. Diff. Eq., 223(2006), no. 1, 161-184. MR 2210142 (2006j:35248)
  • [KLV] J.R. King, A.A. Lacey, and J.L. Vázquez, Persistence of corners in Hele-Shaw flow, Euro. J. Appl. Math., 6(1995), no. 5, 455-490. MR 1363758 (97a:76037)
  • [K] M.K. Korten, Nonnegative solutions of $ u_t = \Delta(u - 1)_{+}:$ Regularity and uniqueness for the Cauchy problem, Nonl. Anal., Th., Meth., and Appl., 27(1996), no. 5, 589-603. MR 1396031 (97h:35089)
  • [KM] M.K. Korten and C.N. Moore, Regularity for solutions of the two-phase Stefan problem, Comm. Pure Appl. Anal. 7 (2008), 591-600. MR 2379443
  • [MR] 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.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 76D27, 35K65, 49J40

Retrieve articles in all journals with MSC (2000): 76D27, 35K65, 49J40

Additional Information

Ivan A. Blank
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Marianne K. Korten
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Charles N. Moore
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Keywords: Mesa problem, Hele-Shaw problem, Stefan problem, free boundary, mushy region, singular limit
Received by editor(s): October 18, 2006
Published electronically: October 10, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society