On Hele-Shaw models with surface tension

Abstract

It is shown that surface tension effects on the free boundary have a regularizing effect for Hele-Shaw models, which implies existence and uniqueness of classicalsolutions for generalinitialdomains. 1. Introduction and main results Recently, N. Alikakos, P. Bates, and X. Chen (1) proved that level sur- faces of solutions to the Cahn-Hilliard equation tend to solutions of the two-phase Hele-Shaw problem with surface tension under the assumption that classical solutions ofthe latter exist. In the present note we are able to guarantee that the above assumption is in fact satisfied. More precisely, our results (11) show the existence ofa unique classical solution to one- and two-phase Hele-Shaw models with surface tension for general initial data. It should be emphasized that even weak solutions to Hele-Shaw models with surface tension were not known to exist in the general setting presented here. In this note we only give the statements ofour results and a brief sketch of their proofs. The full details will appear in (11). We first consider the one-phase problem. Let Ω be a bounded domain in R n and assume that its boundary ∂Ω is ofclass C ∞ . Moreover, assume that ∂Ω consists oftwo disjoint non-empty components J and Γ. Later on, we will model over the exterior componen tΓam oving interface, whereas the interior component J describes a fixed portion ofthe boundary. Let ν denote the outer unit normal field over Γ and fix α ∈ (0, 1). Given a> 0, let