Abstract
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.
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1 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.
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 \(u_{i}\in [0,1]\) is defined as a local volume fraction of a phase \(i\) and related by the constraint \(\sum \limits _{i}^Nu_{i}=1\).
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 \(\varepsilon _{ij}^2=\varepsilon _{ji}^2>0\) and \(a_{ij}=a_{ji}>0\) are the gradient energy coefficient and energy barrier height of the phases \(i\) and \(j\), respectively; \(\lambda >0\) is the coupling constant. The dimensionless temperature \(\theta\) is related to the internal energy \(e\) by
where \(l_i (i=1,\cdots , N)\) are related to the latent heats associated to each kind of physical state. The interpolation functions \(h(u_i)=u_{i}^2(3-2u_i)\) 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, \(\tau _{ij}=\tau _{ji}>0\) 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 \(\sum \limits _{i=1}^{3}u_{i}=1\) and \(\frac{\partial u_i}{\partial u_j}=-1\) for \(i\ne j\).
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 \(\Omega \subset {\mathbb {R}}^n\) is an open bounded domain and \(0<T<\infty\), 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 initial-boundary 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 \(\Omega =(a,b)\) be a bounded open interval with \(a<b\) and \(Q_{T}=(0,T)\times \Omega\) for \(0<T<\infty\). We denote the usual Sobolev spaces for \(1\le p \le +\infty\), \(k\in {\mathbb {N}}\) by
endowed with the norm
For \(p=2\), the Hilbert space \(W^{k,2}(\Omega )=H^k(\Omega )\) is defined by
Let \(B\) be a Banach space and \(1\le p<\infty\), 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 \(\Omega\) and known quantities which varies from line to line.
Next, we state the main result of our article.
Theorem 1.1
Suppose that \(\vert \frac{\varepsilon _{12}^2}{\tau _{12}}-\frac{\varepsilon _{13}^2}{\tau _{13}}\vert\), \(\vert \frac{\varepsilon _{12}^2}{\tau _{12}}-\frac{\varepsilon _{23}^2}{\tau _{23}}\vert\) are sufficiently small, and \(g\in L^2(Q_T)\). Then, for any \((u_{10},u_{20},u_{30})\in [H_0^1(\Omega )]^3\) with \(u_{10}+u_{20}+u_{30}=1\) a.e. in \(\Omega\), and \(\theta _0\in H_0^1(\Omega )\), the initial-boundary value problem (1.3)–(1.4) possesses a global weak solution \((u_1,u_2,u_3,\theta )\in [C([0,T];H_{0}^1(\Omega ))]^4\).
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 that auxiliary problem has global-in-time weak solutions. Finally, we complete the proof of Theorem 1.1.
2 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 \(u_{1t}+u_{2t}+u_{3t}=0\) in \(Q_{T}\), we have \(u_1+u_2+u_3=1\) in \(Q_{T}\). So, by replacing \(u_3=1-u_1-u_2\) in (1.3)–(1.4), we get the following equivalent problem:
where \(k_1=\frac{\varepsilon _{13}^2}{\tau _{13}},k_2=\frac{\varepsilon _{23}^2}{\tau _{23}}, k_3=\frac{\varepsilon _{12}^2}{\tau _{12}},\alpha _{1}=\frac{a_{13}}{\tau _{13}},\alpha _{2}=\frac{a_{23}}{\tau _{23}},\alpha _{3}=\frac{a_{12}}{\tau _{12}}\),
\(\beta _1=\frac{6\lambda l_1}{\tau _{13}}+\frac{6\lambda l_1}{\tau _{12}}-\frac{6\lambda l_3}{\tau _{13}},\beta _{2}=\frac{6\lambda l_2}{\tau _{12}}, \beta _{3}=\frac{6\lambda l_3}{\tau _{13}}, \gamma _1=\frac{6\lambda l_2}{\tau _{21}}+\frac{6\lambda l_2}{\tau _{23}}-\frac{6\lambda l_3}{\tau _{23}}\),
\(\gamma _2=\frac{6\lambda l_1}{\tau _{21}}\), and \(\gamma _3=\frac{6\lambda l_3}{\tau _{23}}\).
It is convenient to rewrite that
Let us define a weak solution for the auxiliary problem (2.1) as follows.
Definition 2.1
Let \(u_{10},u_{20}\in H_0^1(\Omega ),\) and \(\theta _0\in H_0^1(\Omega )\). A function \((u_1,u_2,\theta )\) with
is a weak solution to the auxiliary problem (2.1) for all \(\phi \in C_{0}^{\infty }\left( (-\infty ,T)\times \Omega \right)\) such that
Next, we state the result on the existence of global weak solutions concerning with the auxiliary problem (2.1). Then, by defining \(u_3=1-u_1-u_2\), we obtain a solution \((u_1,u_2,u_3,\theta )\) to the original problem (1.3)–(1.4).
Proposition 2.2
Assume that \((u_{10},u_{20},\theta _{0})\in [H_0^1(\Omega )]^3\) and \(g\in L^2(Q_T)\). If \(\vert k_3-k_1\vert\) and \(\vert k_3-k_2\vert\) are small enough, then the auxiliary problem (2.1) admits at least one global weak solution \((u_1,u_2,\theta )\) in the sense of Definition 2.1.
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-in-time a priori estimates.
2.1 Existence of Local Solutions
Now let us define a nonlinear operator:
where \((u_1,u_2,\theta )\) is the solution of the following auxiliary linear problem
Here \(Q_{t^\star }=(0,t)\times \Omega\) with \(0<t<t^{\star }\).
For this linearized initial-boundary value problem (2.2), we study the existence of local solutions by using the Banach fixed-point theorem [24].
Lemma 2.3
Assume that \((u_{10},u_{20},\theta _{0})\in [H_0^1(\Omega )]^3\) and \(g\in L^2(Q_T)\). If \(\vert k_3-k_1\vert ,\vert k_3-k_2\vert,\) and \(t^{\star }>0\) are small enough, then there exists a unique local solution \((u_1,u_2,\theta )\in [C([0,t^\star ]; H_{0}^1(\Omega ))]^3\) to problem (2.2).
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 \(M>0\):
with norm \(\Vert (u_1,u_2,\theta )\Vert _E=\text {max}\{\Vert u_1\Vert _{E_1},\Vert u_2\Vert _{E_2},\Vert \theta \Vert _{E_3}\}\).
Let us consider \(\varepsilon _0>0\) so small that \(\vert k_3-k_1\vert <\varepsilon _{0}\) and \(\vert k_3-k_2\vert <\varepsilon _0\) then
Observe that, if \(\mu _1,\mu _2\in L^{\infty }(0,t^\star ;H_0^1(\Omega ))\), then owing to the Sobolev embedding \(H_0^1(\Omega )\hookrightarrow L^\infty (\Omega )\), we have
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 \((\mu _1,\mu _2,\vartheta )\in E\), then owing to the Sobolev embedding \(H_0^1(\Omega )\hookrightarrow L^\infty (\Omega )\), the right-hand side of each equation of (2.2) belongs to \(L^{2}(Q_{t^\star })\). Thus for any suitable choice of \(M\), one can apply the theory of linear parabolic equation (see e.g. Evans [26]) such that \((u_1,u_2,\theta ) \in E\), i.e., the operator \(\mathcal {F}\) maps from \(E\) to \(E\).
2. Next, we need to prove that \(\mathcal {F}\) is a contraction, i.e.,
for some \(0 \le \lambda <1\) and all \((\mu _1,\mu _2,\vartheta ),(\mu _1^{\prime },\mu _2^{\prime },\vartheta ^{\prime })\in E\).
Now, for any \((\mu _1,\mu _2,\vartheta ),(\mu _1^{\prime },\mu _2^{\prime },\vartheta ^{\prime })\in E\), we have by considering \(({\tilde{\mu }}_1,{\tilde{\mu }}_2,{\tilde{\vartheta }}) =(\mu _1,\mu _2,\vartheta )-(\mu _1^{\prime },\mu _2^{\prime },\vartheta ^{\prime })\) and \(({\tilde{u}}_1,{\tilde{u}}_2,\tilde{\theta })=(u_1,u_2,\theta )-(u_1^{\prime },u_2^{\prime },\theta ^{\prime })\) such that \(({\tilde{u}}_1,{\tilde{u}}_2,\tilde{\theta })=\) \(\mathcal {F}\) \(({\tilde{\mu }}_1,{\tilde{\mu }}_2, \tilde{\vartheta })\) satisfies
where
Since \((\mu _1,\mu _2, \vartheta ),(\mu _1^{\prime },\mu _2^{\prime }, \vartheta ^{\prime }) \in {L^{\infty }(0,t^\star ;H_0^1(\Omega ))}\), then by \(H_0^1(\Omega )\hookrightarrow L^\infty (\Omega )\), we have \(A_i,B_i\in L^{\infty }(Q_{t^\star })\), for \(i=1,2,3\) and \(D_{i} \in L^{\infty }(Q_{t^\star })\) for \(i=1,\cdots ,6\).
Now, by multiplying the first equation of (2.5) by \(\tilde{u}_1\), integrating in \(\Omega \times (0,t)\) with \(0< t < t^\star\), using Hölder’s inequality, Sobolev embedding, and Young’s inequality, we obtain
By using inequality (2.3), we have \(k_1+(k_3-k_1)\mu _{2}\ge k_{1}/2\) for the second term on the left-hand side of inequality (2.6). Therefore, by choosing any appropriate constant \(\delta >0\), using the definition of norm in \(E\), and applying Gronwall’s lemma for all \(0\le t\le t^\star\), we get
By multiplying the first equation of (2.5) by \({\tilde{u}}_{1t}\), integrating in \(\Omega \times (0,t)\) with \(0< t < t^\star\), using Hölder’s inequality, Sobolev embedding, Young’s inequality, and proceeding in a similar way, we deduce that
The above inequality implies that
Observe that the first equation of problem (2.5) can be written as
Next, by multiplying the equation (2.11) by \(-\tilde{u}_{1xx}\), integrating in \(\Omega \times (0,t)\) with \(0< t <t^\star\), using Hölder’s inequality, Sobolev embedding, and Young’s inequality, we have
Similarly, by using the inequality (2.3), we have \(k_1+(k_3-k_1)\mu _{2}\ge k_{1}/2\) for the second term on the left-hand side of inequality (2.12). Therefore, by choosing any appropriate constants \(\delta , \varepsilon >0\), using the definition of norm in \(E\), utilizing (2.8), and employing standard elliptic estimate, we deduce that
Thus, we conclude that
Proceeding in the same way with the second equation of problem (2.5) and utilizing inequality (2.4), we can also derive that
Thus,
Next, we need to find the necessary estimates for \({\tilde{\theta }}\). So, we will deal with the third equation of problem (2.5).
Therefore, by multiplying the third equation of problem (2.5) by \({{\tilde{\theta }}}\), integrating in \(\Omega \times (0,t)\) with \(0< t<t^\star\), and using Hölder’s and Young’s inequalities, we get
So by choosing appropriate constants \(\delta , \varepsilon >0\), using the definition of norm in \(E\), utilizing (2.13)-(2.14), and applying Gronwall’s lemma, we deduce that
Similarly, by multiplying the third equation of problem (2.5) successively by \(-{{\tilde{\theta }}}_{xx}\) and \({\tilde{\theta }}_t\), integrating in \(\Omega \times (0,t)\) with \(0< t<t^\star\), and proceeding as like the last one, we conclude that
Therefore,
Thus from inequalities (2.13), (2.14), and (2.16), we have that \(\mathcal {F}\) \(:E\rightarrow E\) is a contraction provided \(t^{\star }>0\) so small that \((Ct^\star )^{1/2}=\lambda <1\). Thus, Lemma 3.1 is proved. \(\square\)
3 Uniform a Priori Estimates
In this section, we establish a priori estimates for solutions \((u_1,u_2,\theta )\) of the problem (2.1) for arbitrary \(T>0\).
Lemma 3.1
There exits a constant \(C\) independent of \(t^{\star }\) such that, for any \(T>0,\)
Proof
By differentiating the free energy functional \(F(u_i,u_j,\theta )\) 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 \(t\in (0,T)\), we obtain that
Thus, by using the definition of \(F(u_i,u_j,\theta )\) and initial conditions (1.4)\(_2\) into Eq. (3.2), we can prove Lemma 3.1. \(\square\)
Lemma 3.2
There exits a constant \(C\) independent of \(t^{\star }\) such that, for any \(T>0,\)
Proof
From Lemma 3.1, we observe that \(a.e.\) \(t>0,\)
Thus, we deduce from (3.5) that
and
Then, by inserting \(u_3=1-u_1-u_2\) in estimates (3.7)-(3.8), one yields
Therefore, we apply Minkowski’s inequality for (3.6) and (3.9), and then use Poincaré’s inequality to obtain
From inequality (3.5), we also infer that
Let us recall Lemma 3.1, which also implies that
It is obvious that this inequality provides \(\theta _x\) 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
To estimate \(J\), we use Hölder’s inequality, Sobolev embedding, and estimates \(u_1,u_2\in L^{\infty }(0,T;H_0^1(\Omega ))\) and \(\theta \in L^2(0,T;H_0^1(\Omega ))\) such that
Let \(\eta , \sigma >0\) be small constants such that
Next, we integrate inequality (3.17) over \(\Omega \times (0,T)\) and use the bound of \(J\) to obtain
Thus, it follows after replacing \(I\), that
Similarly, we obtain from (3.15) that
Hence, by using Minkowski’s inequality for (3.18) and (3.19), we get that
By using (3.20) with elliptic estimate, one yields
Proceeding in the previous way, one can also obtain from the second term of left-hand side of (3.12) that
Thus, combining (3.10) with (3.21) and (3.22), estimate (3.3) holds. Hence Lemma 3.2 is proved. \(\square\)
Lemma 3.3
There exits a constant \(C\) independent of \(t^{\star }\) such that, for any \(T>0,\)
Proof
By multiplying the first and second equations of problem (2.1) by \(u_{1t}\) and \(u_{2t}\), respectively, integrating in \(\Omega\), using Hölder’s inequality, Sobolev embedding, Young’s inequality, and adding the resulting inequalities, we have
where the positive constants \(c_{i},\) for \(i=1,\cdots ,6,\) depend on the known parameters.
Now, let us integrate inequality (3.26) over \(0< t< T\), and use the facts that \(u_1,u_2\in {L^{\infty }(0,T;H_0^1(\Omega ))}\) and \(\theta \in {L^{2}(0,T;L^2(\Omega ))}\) such that
By choosing sufficiently small \(\delta ,\varepsilon >0\), we arrive at (3.23). Observe that inequality (3.27) also implies (3.24).
Similarly, by multiplying the third equation of problem (2.1), successively, by \(-\theta _{xx}\) and \(\theta _t\), integrating over \(\Omega \times (0,t)\) with \(0< t< T\), using \(g\in {L^{2}(Q_T)}\), utilizing estimates of \(u_1,\) \(u_2\) from (3.3) and corresponding time derivatives \(u_{1t},\) \(u_{2t}\) from (3.24), we can prove easily (3.25). Thus, the proof of Lemma 3.3 is completed. \(\square\)
Proof
Since Lemma 2.3 provides the existence of local weak solution \((u_1,u_2,\theta )\) to problem (2.1), therefore, Lemmas 3.2-3.3 for uniform a priori estimates allow us to extend the local weak solution \((u_1,u_2,\theta )\) to \([0,T]\) for arbitrary \(T>0\). This means that \((u_1,u_2,\theta )\) is a global weak solution of (2.1). Thus, the proof is established. \(\square\)
Finally, let us use Proposition 2.2 to prove our main result.
Proof of Theorem 1.1
By Proposition 2.2, and using the result of continuity of the embedding \(\{L^2(0,T; H^2(\Omega ))\cap H^1(0,T; L^2(\Omega ))\}\hookrightarrow C([0,T];H_{0}^1(\Omega ))\)(see e.g. Evans [26]), we obtain that \((u_1,u_2,\theta )\in [C([0,T];H_{0}^1(\Omega ))]^3\). Recalling \(u_3=1-u_1-u_2\) implies that \((u_1,u_2,u_3,\theta )\in [C([0,T];H_{0}^1(\Omega ))]^4\) is a global weak solution of (1.3)–(1.4) such that \(u_1+u_2+u_3=1\). Hence, the proof is finished. \(\square\)
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References
Langer, J.S.: Models of pattern formation in first-order phase transitions, Directions in Condensed Matter Physics: Memorial Volume in Honor of Shang-Keng Ma, World Scientific, 1986. https://https://doi.org/10.1142/9789814415309-0005
Caginalp, G., Fife, P.: Phase-field methods for interfacial boundaries, Phys. Rev. B, 33, 7792–7794 (1986). https://doi.org/10.1103/PhysRevB.33.7792
Penrose, O., Fife, P.C.: On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Physica D 69, 107–113 (1993). https://doi.org/10.1016/0167-2789(93)90183-2
Wang, S.L., Sekerka, R.F., Wheeler, A.A., Murray, B.T., Coriell, S.R., Braun, R.J., McFadden, G.B.: Thermodynamically-consistent phase-field models for solidification. Physica D 69, 189–200 (1993). https://doi.org/10.1016/0167-2789(93)90189-8
Wheeler, A.A., Boettinger, W.J., McFadden, G.B.: Phase-field model for isothermal phase transitions in binary alloys. Phys. Rev. A 45, 7424–7439 (1992). https://doi.org/10.1103/PhysRevA.45.7424
Karma, A., Rappel, W.J.: Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57, 4323–4349 (1998). https://doi.org/10.1103/PhysRevE.57.4323
Steinbach, I., Pezzolla, F.: A generalized field method for multiphase transformations using interface fields. Physica D 134, 385–393 (1999). https://doi.org/10.1016/S0167-2789(99)00129-3
Steinbach, I., Pezzolla, F., Nestler, B., Seeßelberg, B.M., Prieler, R., Schmitz, G.J., Rezende, J.L.L.: A phase-field concept for multiphase systems. Physica D 94, 135–147 (1996). https://doi.org/10.1016/0167-2789(95)00298-7
Nestler, B., Wheeler, A.A.: A multi-phase-field model of eutectic and peritectic alloys: numerical simulation of growth structures. Physica D 138, 114–133 (2000). https://doi.org/10.1016/S0167-2789(99)00184-0
Nestler, B., Garcke, H., Stinner, B.: Multicomponent alloy solidification : phase-field modeling and simulations. Phys. Rev. E 71, 041609 (2005). https://doi.org/10.1103/PhysRevE.71.041609
Folch, R., Plapp, M.: Quantitative phase-field modeling of two-phase growth. Phys. Rev. E 72, 011602 (2005). https://doi.org/10.1103/PhysRevE.72.011602
Kim, S.G., Kim, W.T., Suzuki, T., Ode, M.: Phase-field modeling of eutectic solidification. J. Cryst. Growth 261, 135–158 (2004). https://doi.org/10.1007/s11837-004-0070-1
Bollada, P.C., Jimack, P.K., Mullis. A.M.: Multiphase field modelling of alloy solidification. Comput. Mater. Sci., 171, 109085 (2020). https://doi.org/10.1016/j.commatsci.2019.109085
Caginalp, G., Jones, J.: A derivation and analysis of phase-field models of thermal alloys. Ann. Phys. 237, 66–107 (1995). https://doi.org/10.1006/aphy.1995.1004
Caginalp, G., E. Socolovsky, E.: Phase-field computations of single-needle crystals, crystal growth, and motion by mean curvature. SIAM J. Sci. Comput., 15, 106–126 (1994). https://doi.org/10.1137/0915007
Colli, P., Kurima, S.: Global existence for a phase separation system deduced from the entropy balance. Nonlinear Anal.-Theory Methods Appl., 190, 111613 (2020). https://doi.org/10.1016/j.na.2019.111613
Colli, P., Laurenşot, P.: Weak solutions to the Penrose-Fife phase-field model for a class of admissible heat flux laws. Physica D 111, 311–334 (1998). https://doi.org/10.1016/S0167-2789(97)80018-8
Colli, P., Sprekels, J.: Weak solution to some Penrose-Fife phase-field systems with temperature-dependent memory. J. Differ. Equ. 142, 54–77 (1998). https://doi.org/10.1006/jdeq.1997.3344
Hoffman, K.H., Jiang, L.: Optimal control of a phase-field model for solidification. Numer. Funct. Anal. Optim. 13, 11–27 (1992). https://doi.org/10.1080/01630569208816458
Boldrini, J.L., Caretta, B.M.C., Fernandez-Cara, E.: Analysis of a two-phase field model for the solidification of an alloy. J. Math. Anal. Appl. 357, 25–44 (2009). https://doi.org/10.1016/j.jmaa.2009.03.063
Boldrini, J.L., Planas, G.: Weak solutions of a phase-field model for phase change of an alloy with thermal properties. Math. Meth. Appl. Sci. 25, 1177–1193 (2002). https://doi.org/10.1002/mma.334
Alber, H.D., Zhu, P.: Solutions to a model with nonuniformly parabolic terms for phase evolution driven by confrontational forces. SIAM J. Appl. Math. 66, 680–699 (2005). https://doi.org/10.1137/050629951
Alber, H.D., Zhu, P.: Solutions to a model for interface motion by interface diffusion. Proc. R. Soc. Edinb. Sect. A-Math. 138, 923–955 (2008). https://doi.org/10.1017/S0308210507000170
Caretta, B.M.C., Boldrini, J.L.: Local existence of solutions of a three phase-field model for solidification. Math. Meth. Appl. Sci. 32, 1496–1518 (2009). https://doi.org/10.1002/mma.1094
Tang, Y., Gao, W.: Solutions to Three-Phase-Field Model for Solidification. Symmetry 14, 862 (2022). https://doi.org/10.3390/sym14050862
Evans, L.C.: Partial differential equations, American Mathematical Society, 1999
Acknowledgements
The authors would like to thank Professor Peicheng Zhu for his helpful discussions and comments.
Funding
This work was supported by Science and Technology Commission of Shanghai Municipality, PR China under grant no. 20JC1413600.
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Hossain, M.A., Ma, L. Global Weak Solutions to an Initial-Boundary Value Problem for a Three-phase Field Model of Solidification. J Nonlinear Math Phys 30, 475–492 (2023). https://doi.org/10.1007/s44198-022-00081-6
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DOI: https://doi.org/10.1007/s44198-022-00081-6