Hele-Shaw Model for Melting/Freezing with Two Dendrits

Abstract

Melting/freezing process with two dendrits (or freeze “pipes”) is modelled by the complex Hele-Shaw moving boundary value problem in a doubly connected domain. The later is equivalently reduced to a couple of problems, namely, to the linear Riemann-Hilbert boundary value problem in a doubly connected domain and to evolution problem, which can be written in a form of an abstract Cauchy-Kovalevsky problem. The later is studied on the base of Nirenberg-Nishida theorem, and for the former a generalization of the Schwarz Alternation Method is proposed. By using composition of these two approaches we get the local in time solvability of this couple of problems in appropriate Banach space setting.