The Hele–Shaw problem with surface tension in a half-plane

In this paper, we consider the Hele–Shaw problem in a 2-dimensional fluid domain Ω(t) which is constrained to a half-plane. The boundary of Ω(t) consist of two components: Γ0(t) which lies on the boundary of the half-plane, and Γ(t) which lies inside the half-plane. On Γ(t) we impose the classical boundary conditions with surface tension, and on Γ0(t) we prescribe the normal derivative of the fluid pressure. At the point where Γ0(t) and Γ(t) meet, there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure. We prove that the problem has a unique solution with smooth free boundary Γ(t) for some small time interval