Classical Solutions of Multidimensional Hele – Shaw Models

Existence and uniqueness of classical solutions for the multidimensional expanding Hele–Shaw problem are proved. 1. The problem. We are concerned with a class of moving boundary problems for bounded domains in R n , which comprise in particular the so-called single phase Hele–Shaw problem. In order to describe precisely the involved geometry, let Ω be a bounded domain in R n and assume that its boundary ∂Ω is of class C ∞. Moreover, assume that ∂Ω consists of two disjoint nonempty components J and Γ. Later on, we will model over the exterior component Γ a moving interface, whereas the interior component J describes a fixed portion of the boundary. Let ν denote the outer unit normal field over Γ and fix α ∈ (0, 1). Given a > 0, set U := {ρ ∈ C 2+α (Γ) ; ρ C 1 (Γ) < a}. For each ρ ∈ U define the map θ ρ := id Γ + ρν and let Γ ρ := im(θ ρ) denote its image. Obviously, θ ρ is a C 2+α diffeomorphism mapping Γ onto Γ ρ , provided a > 0 is chosen sufficiently small. In addition, we assume that a > 0 is small enough such that Γ ρ and J are disjoint for each ρ ∈ U. Let Ω ρ denote the domain in R n being diffeomorphic to Ω and whose boundary is given by J and Γ ρ. To describe the evolution of the hypersurface Γ ρ , fix T > 0 and set I := [0, T ]. Then each map ρ : I → U defines a collection of domains Ω ρ(t) , t ∈ I. For later purposes it is convenient to introduce the following generalized parabolic cylinder: Ω ρ,T := (x, t) ∈ R n × [0, T ] ; x ∈ Ω ρ(t) = t∈I Ω ρ(t) × {t} and, correspondingly, Γ ρ,T := (x, t) ∈ R n × [0, T ] ; x ∈ Γ ρ(t) = t∈I Γ ρ(t) × {t} .