Local existence of solutions of a three phase‐field model for solidification

Abstract

In this article we discuss the local existence and uniqueness of solutions of a system of parabolic differential partial equations modeling the process of solidification/melting of a certain kind of alloy. This model governs the evolution of the temperature field, as well as the evolution of three phase-field functions; the first two describe two different possible solid crystallization states and the last one describes the liquid state.

Analysis of a two-phase field model for the solidification of an alloy

“…In the present paper, we have chosen to consider this simplification, in order to isolate the mathematical difficulties coming just from the interaction potentials. From the mathematical point of view, the problem in [44] and [45] is harder to analyze; some results in this direction can be found in Caretta and Boldrini [14].…”

Global Weak Solutions to an Initial-Boundary Value Problem for a Three-phase Field Model of Solidification

“…For this linearized initial-boundary value problem (2.2), we study the existence of local solutions by using the Banach fixed-point theorem [24].…”

“…Analytical results for various phase-field models have been studied in Caginalp et al [14,15], Colli et al [16][17][18], Hoffman and Jiang [19], Boldrini et al [20,21], and Alber and Zhu [22,23]. Boldrini et al [24] proved the existence of local solutions to a three-phase field model for one-dimensional case. Recently, Tang and Gao [25] investigated global weak solutions to a three-phase field model…”

Solutions to Three-Phase-Field Model for Solidification

“…Proof. We only analyze Equation (7); Equation ( 8) is treated in a similar way. To prove that u(t, x) ≥ 0, let us define…”

“…[6]). The classic solutions are proved in [7], where the authors set θ = e − l(u + v + w). In this paper, we study the non-isothermal solidification of ideal multi-component and multi-phase alloy systems.…”