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

In this article, we study an initial-boundary value problem for a three-phase field model of nonisothermal solidification processes in the case of two possible crystallization states. The governing equations of the model are the three phase-field equations coupled with a nonlinear heat equation. Each equation of the model has strong nonlinearities involving the higher-order derivatives. We prove the existence of global-in-time weak solutions to our problem for one-dimensional case.


Introduction
Phase-field method is a very powerful computational tool in describing the evolution of microstructure without tracking the interface position. In phase-field method, an order parameter or field variable is used to describe the physical state of various regions or components in a system. Several phase-field models have been successfully developed for material processes, such as solidification, precipitate growth and coarsening, solid-state phase transformations, and crack propagation, etc.
Md Akram Hossain and Li Ma contributed equally to this work.

3
The first phase-field model for solidification of pure material was proposed by Langer [1]. After that, important contributions on development of the phase-field models were made by Caginalp [2], Penrose and Fife [3], Wang and Sekerka [4], Wheeler, Boettinger and McFadden [5], Karma and Rappel [6].
In recent years, the phase-field methodology has been extended to describe the evolution of more than two phases by adopting multiple field variables. The main concept of multiphase-field system lies on the work of Steinbach et al. [7,8]. Later, several multiphase systems were developed by Nestler et al. [9,10], Folch and Plapp [11], Kim et al. [12], and Bollada et al. [13].
In the standard multiphase-field system for N-phases, each phase-field u i ∈ [0, 1] is defined as a local volume fraction of a phase i and related by the constraint In this article, we study a multiphase-field model based on the Steinbach et al. in [8]. In order to get our multiphase-field model, we consider the following energy functional where 2 ij = 2 ji > 0 and a ij = a ji > 0 are the gradient energy coefficient and energy barrier height of the phases i and j , respectively; > 0 is the coupling constant. The dimensionless temperature is related to the internal energy e by where l i (i = 1, ⋯ , N) are related to the latent heats associated to each kind of physical state. The interpolation functions h(u i ) = u 2 i (3 − 2u i ) satisfy h(u i ) = 1 at u i = 1 and h(u i ) = 0 at u i = 0.
The dynamics of the phases u i are derived by the minimization of the free energy functional F, Here, ij = ji > 0 is a relaxation time at an interface between i and j phases, and we assumed that in the triple point the transition between the phases occurs by the movement of the dual phase boundaries, which do not influence each other [8].
For a system of three-phases, we consider u 1 , u 2 , and u 3 are three phase-field variables that represent the solid fractions of the two possible different kinds of crystallization states and the liquid state, respectively, such that Thus, we have the following PDE system from the dynamic equation (1.2) for three-phases coupling with an energy equation, Here, the function g is related to the density of heat sources or sinks and the given constant b > 0 stands for thermal conductivity. We consider the system (1.3) equipped with the following initial and Dirichlet's boundary conditions where Ω ⊂ ℝ n is an open bounded domain and 0 < T < ∞ , and the initial conditions of phase-field variables satisfy u 10 + u 20 + u 30 = 1. Our aforementioned mathematical model (1.3)-(1.4) plays a vital role in describing the complex growth phenomena during solidification or melting of certain metallic alloys in which two different kinds of crystallization are possible [7][8][9].
In this article, we will study the existence of global weak solutions for the initialboundary value problem (1.3)-(1.4) in one-dimensional domain. The problem has very strong nonlinearities involving the higher-order derivatives. The mathematical analysis of such a model is much more difficult than any single phase-field model.
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 of solidification where they assumed the variable co-efficients of highest order derivative terms in each equation as positive constants. To our knowledge, there are a few theoretical results available for multi-phase systems. We first introduce some notations and then formulate the main result. Let us consider that all functions depend only on the variables x 1 and t . To simplify the notation, we write x 1 by x . Let Ω = (a, b) be a bounded open interval with a < b and Q T = (0, T) × Ω for 0 < T < ∞ . We denote the usual Sobolev spaces for 1 ≤ p ≤ +∞ , k ∈ ℕ by endowed with the norm For p = 2 , the Hilbert space W k,2 (Ω) = H k (Ω) is defined by Let B be a Banach space and 1 ≤ p < ∞ , we consider the functional spaces by We will frequently use the following result from the Sobolev embedding that for some constant C > 0, Throughout this article, we denote C as a generic positive constant depending on Ω and known quantities which varies from line to line.
Next, we state the main result of our article. The remaining parts of this work are devoted to the proof of Theorem 1.1. In Section 2, we formulate an auxiliary problem (2.1) by replacing u 3 = 1 − u 1 − u 2 in problem (1.3)-(1.4) and find local weak solutions of that auxiliary problem by using fixed-point argument. In Section 3, we derive uniform a priori estimates to deduce , for any f ∈ H 1 0 (Ω).
that auxiliary problem has global-in-time weak solutions. Finally, we complete the proof of Theorem 1.1.

An Auxiliary Problem
First, we need to consider an auxiliary problem to obtain the existence of solutions to problem (1.3)-(1.4). Owing to the condition u 10 + u 20 + u 30 = 1 and It is convenient to rewrite that Let us define a weak solution for the auxiliary problem (2.1) as follows.
is a weak solution to the auxiliary problem (2.1) for all ∈ C ∞ 0 ((−∞, T) × Ω) such that Next, we state the result on the existence of global weak solutions concerning with the auxiliary problem (2.1). Then, by defining The proof of Proposition 2.2 consists of a couple of steps. First, we linearize the problem (2.1) to obtain a unique local solution. Then, we use the method of continuation of local solutions to obtain global solutions after deriving some uniform-intime a priori estimates.

Existence of Local Solutions
Now let us define a nonlinear operator: where (u 1 , u 2 , ) is the solution of the following auxiliary linear problem

3
Journal of Nonlinear Mathematical Physics (2023) 30:475-492 For this linearized initial-boundary value problem (2.2), we study the existence of local solutions by using the Banach fixed-point theorem [24].
Proof First, we consider the following Banach spaces for i = 1, 2, 3: with norm Next, we will apply the Banach fixed-point theorem on the following closed ball for M > 0: , then owing to the Sobolev embedding (2.3) Later, the inequalities (2.3) and (2.4) will be found very useful to deal with some divergence related terms to the left-hand side of first and second equations of our auxiliary linear problem (2.2). 1. Since ( 1 , 2 , ) ∈ E , then owing to the Sobolev embedding H 1 0 (Ω) ↪ L ∞ (Ω) , the right-hand side of each equation of (2.2) belongs to L 2 (Q t ⋆ ) . Thus for any suitable choice of M , one can apply the theory of linear parabolic equation (see e.g. Evans [26]) such that (u 1 , u 2 , ) ∈ E , i.e., the operator F maps from E to E.

Uniform a Priori Estimates
In this section, we establish a priori estimates for solutions (u 1 , u 2 , ) of the problem (2.1) for arbitrary T > 0.

Lemma 3.1
There exits a constant C independent of t ⋆ such that, for any T > 0, Proof By differentiating the free energy functional F(u i , u j , ) in (1.1) with respect to t , we obtain Considering the dual phase interactions at the interfaces and applying the dynamic equation (1.2), it follows that By integrating (3.1) in time t ∈ (0, T) , we obtain that (3.1) Thus, by using the definition of F(u i , u j , ) and initial conditions ( (3.6) ‖u 2 u 1x − u 1 u 2x ‖ L ∞ (0,T;L 2 (Ω)) ≤ C, Let us recall Lemma 3.1, which also implies that It is obvious that this inequality provides x is bounded in L 2 (Q T ) . Thus, by using Poincaré's inequality, we have Hence, by combining (3.11) together with (3.13), we arrive at (3.4).
Observe that the first three terms on the left-hand side of inequality (3.12) have the same structure. So, we will deal with only first term and others can be dealt with the same way. Now, the first term of inequality (3.12) implies that The last inequality means that and Let us write inequality (3.14) in the following form where (3.11) ‖ ‖ L ∞ (0,T;L 2 (Ω)) ≤ C. (3.12) (3.14) To estimate J , we use Hölder's inequality, Sobolev embedding, and estimates u 1 , u 2 ∈ L ∞ (0, T;H 1 0 (Ω)) and ∈ L 2 (0, T;H 1 0 (Ω)) such that Let , > 0 be small constants such that (3.17) (I + J) 2 + J 2 ≥ I 2 .