A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane
In [7] Richardson derived a mathematical model for describing Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. This model can be represented in the following form (see also [3]): Given fo(z), f0(0)=0, analytic and univalent in a neighbourhood of Izl_<l, find f(z,t), analytic and univalent as a function of z in a neighbourhood of Izl_<l, continuously differentiable with respect to t in a right-sided neighbourhood of t=0 , satisfying