Analytic solutions for a Stefan problem with Gibbs-Thomson correction

Abstract

Abstract. We provide existence of a unique smooth solution for a class of one-and two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the moving interface depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal L pregularity for the linearized problem, which is fully developed in this paper as well.

Existence of analytic solutions for the classical Stefan problem

“…We refer to [27] for more information and results. Finally, we want to remark that this paper also serves as a preparation to a forthcoming paper of the authors on singular limits for the two-phase Stefan problem.…”

Well-posedness of a Two-phase Flow with Soluble Surfactant

“…In fact, the arguments are very similar to those employed in [12] for the Stefan problem with surface tension, and in [14] for the free boundary value problem of the Navier-Stokes equations.…”

“…Concerning the proof, notice first that the "only if"-part follows by taking traces; see section 5 in [12]. For the "if"-part observe that problem (13)- (15) consists of two decoupled subsystems.…”

“…The norm of L −1 is bounded by some constant M , which does not depend on T ∈ (0, T 0 ], where T 0 < ∞ is fixed. We now define H : Z T → X T × Y T by the right-hand side of (11) and by the last boundary condition in (12). Then the nonlinear problem can be rewritten as…”

“…For a general reference to this subject, we refer to the monograph [11]. Of particular importance for this work are the papers [12,13,14,22,23], where basic notations used throughout this paper can be found.…”