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The one-phase Hele-Shaw problem with singularities. (English) Zbl 1155.35496
Summary: In this article we analyze viscosity solutions of the one phase Hele-Shaw problem in the plane and the corresponding free boundaries near a singularity. We find, up to order of magnitude, the speed at which the free boundary moves starting from a wedge, cusp, or finger-type singularity. Maximum principle-type arguments play a key role in the analysis.

MSC:
 35R35 Free boundary problems for PDEs 35Q35 PDEs in connection with fluid mechanics 76D27 Other free boundary flows; Hele-Shaw flows
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References:
 [1] Athanasopoulos, I., Caffarelli, L., and Salsa, S. Regularity of the free boundary in parabolic phase-transition problems,Acta Math. 176, 245–282, (1996). · Zbl 0891.35164 [2] Caffarelli, L. A. Harnack inequality approach to the regularity of free boundaries, Part II: Flat free boundaries are Lipschitz,Comm. Pure Appl. Math. 42, 55–78, (1989). · Zbl 0676.35086 [3] Caffarelli, L. A. Harnack inequality approach to the regularity of free boundaries, Part I: Lipschitz free boundaries are C1,{$$\alpha$$},Rev. Mat. Iberoamericana 3(2), 139–162 (1987). · Zbl 0676.35085 [4] Dahlberg, B. E. Harmonic functions in Lipschitz domains, Harmonic analysis in Euclidean spaces, Part 1, 313–322,Proc. Sympos. Pure Math., Part XXXV, Amer. Math. Soc., Providence, R.I., (1979). · Zbl 0425.31009 [5] Howison, S. D. Cusp development in Hele-Shaw flow with a free surface,SIAM J. Appl. Math. 46, 20–26, (1986). · Zbl 0592.76042 [6] Kim, I. Uniqueness and Existence result of Hele-Shaw and Stefan problem,Arch. Rat. Meck Anal. 168, 299–328, (2003). · Zbl 1044.76019 [7] Kim, I. Regularity of free boundary in one phase Hele-Shaw problem,J. Differential Equations, to appear. · Zbl 1087.76024 [8] Kim, I. Long time regularity of Hele-Shaw and Stefan problem, submitted. · Zbl 1099.35098 [9] King, J. R., Lacey, A. A., and Vazquez, J. L. Persistence of corners in free boundaries in Hele-Shaw flow,European J. Appl. Math. 6, 455–490, (1995). · Zbl 0840.76016
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