# zbMATH — the first resource for mathematics

Regularity of the free boundary for the one phase Hele–Shaw problem. (English) Zbl 1087.76024
Summary: We prove that, in a local neighborhood, Lipschitz continuous free boundary of a solution of the one-phase Hele-Shaw problem is indeed smooth if the solution is Lipschitz continuous and non-degenerate in the neighborhood.

##### MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D27 Other free boundary flows; Hele-Shaw flows 35Q35 PDEs in connection with fluid mechanics
##### Keywords:
Lipschitz continuous boundary
Full Text:
##### References:
 [1] Angenent, S.B., Nonlinear analytic semiflows, Proc. roy. soc. Edinburgh, 115A, 91-107, (1990) · Zbl 0723.34047 [2] Antontsev, S.N.; Gonalves, C.R.; Meirmanov, A.M., Local existence of classical solutions to the well-posed hele – shaw problem, Port. math. (N.S.), 59, 435-452, (2002) · Zbl 1037.35100 [3] Athanasopoulos, I.; Caffarelli, L.; Salsa, S., Regularity of the free boundary in parabolic phase-transition problems, Acta math., 176, 245-282, (1996) · Zbl 0891.35164 [4] Caffarelli, L., The regularity of free boundaries in higher dimension, Acta math., 139, 155-184, (1977) [5] Caffarelli, L., A Harnack inequality approach to the regularity of free boundaries, part I: Lipschitz free boundaries are $$C^{1, \alpha}$$, Rev. mat. iberoamericana, 3, 2, 139-162, (1987) · Zbl 0676.35085 [6] 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 [7] S. Choi, D. Jerison, I. Kim, Regularity for one-phase Hele-Shaw problem from a Lipschitz initial surface, submitted. · Zbl 1189.35384 [8] B.E. Dahlberg, Harmonic functions in Lipschitz domains, Harmonic analysis in Euclidean Spaces, Part 1, pp. 313-322, Proceedings of the Symposium Pure Mathematics, vol. XXXV, Part, American Mathematical Society, Providence, RI, 1979. [9] Daskalopoulos, P.; Lee, K., All time smooth solutions of the one-phase Stefan problem and the hele – shaw flow, Comm. partial differential equations, 29, 71-89, (2004) · Zbl 1099.35175 [10] Elliot, C.M.; Janovsky, V., A variational inequality approach to hele – shaw flow with a moving boundary, Proc. roy. soc. Edinburgh sect. A, 88, 1-2, 93-107, (1981) · Zbl 0455.76043 [11] Escher, J.; Simonett, G., Classical solutions of multidimensional hele – shaw models, SIAM J. math. anal., 28, 5, 1028-1047, (1997) · Zbl 0888.35142 [12] Friedman, A., Variational principles and free boundary problems, pure and applied mathematics, (1982), Wiley New York [13] Jerison, D.S.; Kenig, C.E., Boundary value problems on Lipschitz domains, (), 1-68 [14] Kim, I.C., Uniqueness and existence result of hele – shaw and Stefan problem, Arch. rational mech. anal., 168, 299-328, (2003) · Zbl 1044.76019 [15] I.C. Kim, Long time regularity of Hele-Shaw problem, submitted. · Zbl 1099.35098 [16] Kinderlehrer, D.; Nirenberg, L., Hodograph methods and the smoothness of the free boundary in the one phase Stefan problem, (), 57-69 [17] Meirmanov, A.M.; Zaltzman, B., Global in time solution to the hele – shaw problem with a change of topology, European J. appl. math., 13, 431-447, (2002) · Zbl 1068.76022 [18] Widman, K.-O., Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. scand., 21, 17-37, (1967) · Zbl 0164.13101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.