Theorem on the existence of solutions of quasi-static moving boundary problems

Using the theory of conformal mappings, we show that two-dimensional quasi-static moving boundary problems can be described by a non-linear Löwner-Kufarev equation and a functional relation between the shape of the boundary and the velocity at the boundary. Together with the initial data, this leads to an initial value problem. Assuming that satisfies certain conditions, we prove a theorem stating that this initial value problem has a local solution in time. The proof is based on some straightforward estimates on solutions of Löwner-Kufarev equations and an iteration technique