PORTUGALIAEMATHEMATICA
Vol. 59, No. 4, pp. 435-452 (2002)
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PORTUGALIAE MATHEMATICA Vol. 59, No. 4, pp. 435-452 (2002) |
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Local Existence of Classical Solutions to the Well-Posed Hele--Shaw ProblemS.N. Antontsev, C.R. Gonçalves and A.M. MeirmanovDep. Matemática, Universidade da Beira Interior,R. Marquês D'Ávila e Bolama, 6201-001 Covilha -- PORTUGAL E-mail: anton@ubi.pt Dep. Matemática, Instituto Politécnico da Guarda -- ESTG, Av. Dr. Francisco Sá Carneiro, 50, 6301-559 Guarda -- PORTUGAL E-mail: crg@ipg.pt Dep. Matemática, Universidade da Beira Interior, R. Marquês D'Ávila e Bolama, 6201-001 Covilha -- PORTUGAL E-mail: meirman@noe.ubi.pt Abstract: We prove local existence of classical solutions to the well-posed Hele--Shaw problem under general conditions on the fixed boundaries. Our approach consists of a construction of approximate solutions as the solutions to the one-phase Stefan problem with $\varepsilon$-heat capacity and energy estimates in Von Mises variables. These estimates permit us to find some small time interval where norms of approximate solutions in some Sobolev spaces are bounded and pass to the limit when $\varepsilon$ goes to zero. Keywords: Nonlinear partial differential equations; free boundary problems. Classification (MSC2000): 35-02, 35R35, 76-02, 76D27. Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.
© 2002 Sociedade Portuguesa de Matemática
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