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

Abstract

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.