Solutions to a phase-field model of sea ice growth

*Correspondence: tangyangxin@i.shu.edu.cn 1Department of Mathematics, Shanghai University, Shanghai, P.R. China Abstract We shall apply the phase-field method to investigate the dynamics of sea ice growth. The model consists of two parabolic equations. The existence and uniqueness of weak solutions to an initial-boundary value problem of this model is proved. Then the regularity, large-time behavior of solutions are studied, also the existence of global attractor is proved. The main technique in this article is energy method. Our existence proof is only valid in one space dimension.


Introduction
Due to global warming, which leads to significant climate changes and more and more frequently occurring severe weather disasters, the study of global warming seems more important than ever since sea ice has begun to melt (see, e.g., [1][2][3]) and this makes sea level rise considerably so that some islandish countries may vanish. In this paper we shall employ a phase-field approach to model the growth of sea ice. This approach, though it has been developed since 1980s, thus is very young, is now a very powerful tool for both theoretical and numerical studies in many fields (see, e.g., [4,5]). To our knowledge, the application of a two-phase field model to investigation of sea ice growth presented in this paper is the first one in phase-field modeling for sea ice evolution.
The evolution of macroscopic sea ice has been studied by means of the classic Stefan's problem (see, e.g., [6]). Fluid flow through sea ice is another point of interest. The permeability of sea ice is important in many physical processes such as the melting and draining from sea ice surface during the melting season (see, e.g., [7]). A new simple model which includes turbulent transport of heat and salt between ice and ocean is introduced and solved analytically (see, e.g., [8]). The mesoscopic numerical simulation of sea ice crystals growth has been studied through Voronoi dynamics during the freezing season (see, e.g., [9]). Overall, sea ice interacts with the climate system of the polar. A one-dimensional enthalpy-based model of sea ice allows for quantitative studies of sea ice and its interaction (see, e.g., [10]). These references need to add appropriate conditions at the interface of tracking movement. Theoretical analysis and numerical simulation are very difficult. In this paper we study a phase-field model for the evolution of the phase interface region in sea water-ice interface phase change problems, which was derived in [5]. The author only simulated dendritic crystal growth without theoretical analysis. We will use the phasefield method in the sea ice growth, more precisely, I do theoretical analysis, regularities, and large time behavior. We formulate this initial-boundary value problem in the onedimensional case and conclude the introduction by stating our main result.
Let Ω ⊂ R 3 be an open set. We introduce a phase-field variable (the order parameter p ∈ R) to represent the physical state of the system in time and space, that is, to distinguish the liquid phase and solid phase, such as the solid state when the variable is 1. The liquid phase is expressed when the variable is 0. We restrict ourselves to that type of order parameter p, which describes the evolution of phase interfacial region. Now let us establish the free energy function F of the system based on the order parameter p, their spatial derivatives ∇p, and the local temperature: we choose for ψ ∈ C 2 (R, [0, ∞)) a direct extension of the double well potential with minima at p = 0 and p = 1. Here, e 0 , ε 12 , a 12 , m 12 are thermophysical data. T is temperature, it satisfies where h(p) is a non-decreasing smooth function satisfying h(0) = 0 near p = 0 and h(1) = 1 near p = 1, e is the local enthalpy, and e 0 satisfies L is the latent heat of fusion for sea-water, T M is the melting temperature, and c p * is the specific heat of sea water. For the two-phase case, we get the following systems: for (t, x) ∈ (0, ∞) × Ω, κ, D are constants. The boundary and initial conditions are Now we make some assumptions. We assume that all functions depend on the variables x 1 and t and, to simplify the notation, denote x 1 by x. The set Ω is a bounded open interval. We write Q T e := (0, T e ) × Ω, where T e is a positive constant, and define Then, under these assumptions, equations (1.2)-(1.3) in the case of one dimension can be rewritten as follows: where κ = 2e 0 τ 12 . The boundary and initial conditions therefore are (1.13) The main results of this article are as follows.
for every t, s ≥ 0 (the semigroup property).

Theorem 1.3
Let Ω denote an open bounded set of R and g 1 denote a polynomial. The semigroup p(t) associated with the initial-boundary-value problem (1.8)-(1.13) possesses a maximal attractor A which is bounded in H 1 0 (Ω), compact and connected in L 2 (Ω). Its basin of attraction is the whole space L 2 (Ω), A attracts the bounded sets of L 2 (Ω). Assume that the coefficient is suitably large. Then p L ∞ (Ω) and T 2 decrease exponentially to 0 as t → ∞.

Notation
Let Ω be a domain in R n , and let r be a positive real number. We denote by L p (Ω) the class of all measurable functions u defined on Ω for which where k is any positive integer and D α u is the weak partial derivative. · , | · |, C, A, S(t) denote L 2 (Ω)-norm, the absolute value, various constants, attractor, semigroup, respectively; ∂ t or d dt or a subscript t and ∂ x or a subscript x denote the derivative with respect to t and x in the distribution sense, respectively.
The remaining sections are devoted to the proof of Theorem 1.1, Theorem 1.2, and Theorem 1.3. In order to obtain the local solution of the initial-boundary value problem for nonlinear equations (1.8)-(1.13), we construct the approximate sequence We prove the existence of weak solutions by iterative method: Choose a known approximate solution p n-1 , T n-1 and determine the next p n , T n by solving equations to (1.8)-(1.9), proving existence solution by using of Banach's fixed point theorem. When we regard the term p n in (1.9) as known by use of solution of (1.8), a solution will be obtained if convergence of this procedure can be shown. In Sect. 2 we shall establish some a priori estimates for the solution.
We will discuss the regularity of our weak solutions p, T for the parabolic systems in Sect. 3. Section 4 is devoted to investigation of the large time behavior of a solution by using the a priori estimates.

A priori estimates
In this section we establish a priori estimates for solutions (p, T) to the initial-boundary value problem (1.8)-(1.13).

Lemma 2.2 There holds, for any t
Proof Noting that f | ∂Ω = 0, we have the Gagliardo-Nirenberg inequality in the form We have Taking the L 2 (Ω)-norm on both sides of equation (2.7), squaring and integrating it in τ ∈ (0, t), using relation (2.1), we have Next we invoke the inequality It remains to show that T t ∈ L 2 (0, t; H -1 (Ω)). To do so, (2.8) is changed to and T t is bounded in L 2 (0, t; H -1 (Ω)).
We have the solution p, T of (1.8), (1.9). This allows us to extend the solution p, T stepby-step to all of T e .  (1.16), ϕ = w 2 in (1.17) and integrating by parts, by using Young's inequality, we have By using of Gronwall's inequality, we obtain p = p for almost everywhere Q T e . (2.11) is changed to we obtain T = T for almost everywhere Q T e .

Regularity
In this section we discuss the regularity of the weak solutions p, T to the initial-boundary value problem for parabolic-parabolic systems in Sect. 1. Assume that all conditions in Sect. 1 are met and p 0 ∈ H 2 (Ω), T 0 ∈ H 1 0 (Ω). For this initial p 0 ∈ H 2 (Ω), T 0 ∈ H 1 0 (Ω), solutions p, T can be constructed as in Sect. 2. Our eventual goal is to prove that p, T is smooth.

Global attractor
In this section we discuss the existence of a global attractor and the stability of solution to problem (1.8)-(1.13). This amounts to proving that the solutions of the evolution problem remain bounded as t → ∞. Usually, proving the existence of absorbing sets amounts to proving a priori estimates. Once the properties of the semigroup are established, we may apply the general results of the attractor. That theorem produces the existence of an attractor which is maximal among the bounded attractors and among the bounded functional invariant sets; it fully describes the long-time behavior of the solutions of the equations.  There exists an absorbing set B 0 in L 2 (Ω), namely, any ball of L 2 (Ω) centered at 0 of radius ρ 0 > ρ 0 . If B is a bounded set of L 2 (Ω), included in a ball B(0, R) of L 2 (Ω) centered at 0 of radius R, then S(t)B ⊂ B(0, ρ 0 ) for t ≥ t 0 (B; ρ 0 ) t 0 = 1 log R 2 (ρ 0 ) 2ρ 2 0 .

Global attractor
(4.7) We also infer from (4.4), after integration in t, that (b) Absorbing set in H 1 0 (Ω) of p. We now prove the existence of an absorbing set in H 1 0 (Ω) and the uniform compactness of S(t). For that purpose we need another energytype equality; it is obtained by multiplying (1.8) by -p xx and integrating by parts with