The vanishing surface tension limit for the Hele-Shaw problem

Hyung Ju  Hwang; Youngmin  Oh; Marco Antonio  Fontelos

doi:10.3934/dcdsb.2016108

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2016, Volume 21, Issue 10: 3479-3514. Doi: 10.3934/dcdsb.2016108

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The vanishing surface tension limit for the Hele-Shaw problem

1.
Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784
2.
Instituto de Ciencias Matemáticas (ICMAT), C/Nicolás Cabrera, 28049 Madrid

Received: June 30, 2015

Revised: July 31, 2016

Published: November 2016

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Abstract

In this paper, we show the existence of solutions of the Hele-Shaw problem in two dimensions in the presence of surface tension and for a general class of initial data. The limit problem from nonzero to zero surface tension will also be investigated. In the case of injection and when volume conservation holds, for a sufficiently small surface tension, we prove the existence and uniqueness of perturbed solutions with nonzero surface tension near solutions with zero surface tension. We also show that solutions with nonzero surface tension exist up to a finite time before a possible singularity occurs in which solutions with zero surface tension are well defined. In addition, in the finite time interval, we prove that the solutions with nonzero surface tension approach the solutions with zero surface tension as the surface tension coefficient goes to zero. In the case of suction, for sufficiently small surface tension, we prove the existence of perturbed solutions near solutions with zero surface tension in any initially smooth domains. In this case, the local existence time depends on the surface tension coefficient.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 35B20, 35B25.

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