Weak Solutions to an Initial-Boundary Value Problem of a Consistent Three-phase-field Model

Global warming has caused the melting of glaciers, especially the Greenland sea ice which is a great threat to human beings. The phase-field method is used to describe the evolution of multi-phase microstructure of Greenland sea ice. This paper shall investigate the global existence, uniqueness and regularity of weak solutions to an initial-boundary value problem for a three-phase-field model based on criteria that lead to both physical and mathematical consistency. For global solutions, it is calculated by applying the method of continuation of local solutions. This model allows for the formulation of spurious phases, which is completely consistent with physical reality.


Introduction
Global warming has dramatically changed the current world climate pattern.For instance, it accelerates the melting of the largest glacier area: the Greenland sea ice which is a serious threat to the survival of all human beings.In 2020, Briner, et al. [1] established a high-resolution ice-sheet model to reveal unprecedented large-scale melting in this century.It is meaningful to study the evolution of sea ice in Greenland.
The process of melting of sea ice which is a complex system involves several components, i.e., temperature, sea water, snow, sea ice I, sea ice II and so on.This paper considers it as a whole and employ a multi-phase-field method to simulate the evolution of sea ice.Steinbach, et al [2], in 1966, first established the multi-phase free energy functional for multi-phase systems.For the phase-field method, We refer, such as, to [3][4][5][6].In addition, Tóth and Pusztai [7] proposed seven multi-phase-field criteria to describe a generalized multi-phase-field model for an arbitrary number of phases (or domains) in 2015.There did not exist spurious phases.
Tang [8][9] investigated the well-posedness of global weak solution for a two-phase-field model which does not directly show an expression for temperature and a three-phase-field model with different methods, respectively.While Akram [10] showed an expression for temperature and discussed the global in-time weak solution to a three-phase-field model that describes the evolution of sea ice in one-dimensional case.In their three-phase-field models, there's no spurious phase.However, from a physical point of view, with changing temperatures, sea water evolves into sea ice and snow, the model needs to appear spurious phases, that is, if a phase is absent at the beginning, it may appear doubtlessly in the following time.
Here, the consistent phase-field method is employed to describe the evolution of multi-phase microstructure of Greenland sea ice, moreover, this evolution can be regarded as multi-phase transformations at the microscopic scale.
We mainly study the symmetrical interactions during the three phases: sea water, snow, sea ice and temperature.The model subjected to Neumann boundary condition is: here, , T 0  Ω⸦R 3 is a bounded open region, n denotes the outer unitary normal vector of ∂Ω,  , e D are assumed to be positive constants depending on involved sea ice.The constants r1, r2, r3 are related to the sea ice system.The function f(t,x) is related to the density of sources or sinks of sea ice system.u4 is the temperature, while the phase-field functions u1, u2 and u3 are the respective fractions of the liquid phase field sea water and two different possible solid crystallization phase fields sea ice, snow, thus, physically we must have u1+u2+u3=1.The initial conditions u10, u20, u30, u40 are given suitable functions and u10+u20+u30 =1.To derive this model, the free energy functional is constructed as where here e0 is constant.We use variational methods and the general form of the thermodynamical equations: (kij is a given independent parameter) obtain the multi-phase-field equations constructed to guarantee the second law of thermodynamics.In consistent a priori estimates, straight-forward computations show that.In addition, we choose a specific form of temperature considered as a linear combination of three phase fields, that is, we mainly study the symmetrical interactions during the three phases: sea water, snow, sea ice and temperature.

Main Results
To state the global well-posedness and regularity results for problem (1), we introduce some notations and definition of weak solution first.
In this article, C stands for variable positive constant which will change in different places and only depends on given parameters but not the functions to be estimated throughout this paper.


Then the definition of weak solution is given: is a weak solution to the provided IBVP (1), if the last two equations of (1) are satisfied weakly and if for all test functions Now, we state our main theorems, whose proof will be divided into different sections in the following.
Theorem 1 Suppose an open bounded region

  H u
and for any positive time T, , on the conditions that we choose a suitable there exists a unique global solution for the IBVP (1) under some assumptions.

Consistent a Priori Estimates
In this section, we show the global-in-time weak solutions for the IBVP (1) by applying the method of continuation of local solutions, while the local solution is obviously obtained by standard argument.
Firstly, dealing with the three-phase field free energy functional F given in (2), the free energy decreases and some estimates hold.The estimates results are as follows Lemma 1 For any positive T, the weak solutions of problem (1) Next, we prove the estimates of Multiplying the fourth equation ( 4) by and integrating over T Q and using Hölder and Young's inequalities to calculate here, the fact for any positive T.
By multiplying the fourth equation in (1) by t u   4 , integrating over T Q and using Hölder, Young's inequalities to calculate Summarizing, based on Lemma 1 and the estimates of , u 4 we calculate .Then global solutions for the system (1) is proved.

Uniqueness of Global Solutions Lemma 2
The uniqueness of the global solutions for problem (1) is proved under some conditions.
For the uniqueness, we give a simple proof procedure.For convenience, we reduce (1).Since where Similarly, there exists a constant C such that then we suppose that there exists a suitable constant m such that when Multiplying the first three equations of (9) by , u ~1 2 u ~and 4 u ~respectively, integrating over T Q and using integrating by parts, Hölder, Young's inequalities, similarly, we sum up and obtain here, the fact and the assumptions ( 10)- (11)  Q .The uniqueness of the global solutions is thus proved.That is, Lemma 2 is proved.Moreover, combining the results of Chapter 3, Theorem 1 is completely proved.

Regularity
In this section, the regularity of the weak solutions for problem (1) is established under some conditions.We also deal with problem (8) and give the regularity result 0 ( ( for the IBVP (1) under some assumptions.Proof.To obtain the regularity, we prove it in three steps.Firstly of all, differentiate the first three equations of (8) with respect to t and set Final, differentiate (14) with respect to x to obtain

Lemma 4
Under the hypothesis (10)-(11) and owing to the fact the model by moving the second term of the first three equations of (8) to the right, we arrive at method similarly to deal with (14), we calculate the estimates of high order derivation.
are used.