Advances in Differential Equations

Exact estimates for the classical solutions to the free-boundary problem in the Hele-Shaw cell

Stanislav Antontsev, César Gonçalves, and Anvarbek Meirmanov

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We consider the classical solutions to the multi-dimensional free-boundary problem in the Hele-Shaw cell with general boundary conditions on a given boundary. Exact regularity estimates in the Hölder space are established using the explicit form of the solution to the model linear problem in the half-space and a method for evaluating the convolution integrals. This method was suggested by V.~Solonnikov and is based on the use of Golovkin's theorem. We prove that if the free boundary $\Gamma ( t ) $ is initially $C^{l}$-regular$~(l>2$ is noninteger), then it preserves the same regularity ($\Gamma ( t ) \in C^{l})$ till some instant $T_{\ast}$ depending on the $C^{2}$-norm of the free boundary $\Gamma ( t ) $ and on the topology of $\Gamma ( t ) $. At this instant $T_{\ast}$, either the $C^{2}$-norm of the free boundary $\Gamma ( t ) $ tends to infinity or $\Gamma ( t ) $ changes its topology.

Article information

Adv. Differential Equations Volume 8, Number 10 (2003), 1259-1280.

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R35: Free boundary problems
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J60: Nonlinear elliptic equations 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D27: Other free-boundary flows; Hele-Shaw flows


Antontsev, Stanislav; Gonçalves, César; Meirmanov, Anvarbek. Exact estimates for the classical solutions to the free-boundary problem in the Hele-Shaw cell. Adv. Differential Equations 8 (2003), no. 10, 1259--1280.

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