Theorem on the Existence of Solutions of Quasi-Static Moving Boundary Problems

摘要

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