{'fraction_non_alphanumeric': 0.07303396820983364, 'fraction_numerical': 0.02697066592520506, 'mean_word_length': 4.03734827264239, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Well-posedness of Hibler's dynamical sea-ice model

This paper establishes the local-in-time well-posedness of solutions to an approximating system constructed by mildly regularizing the dynamical sea-ice model of {\it W.D. Hibler, Journal of Physical Oceanography, 1979}. Our choice of regularization has been carefully designed, prompted by physical considerations, to retain the original coupled hyperbolic-parabolic character of Hibler's model. Various regularized versions of this model have been used widely for the numerical simulation of the circulation and thickness of the Arctic ice cover. However, due to the singularity in the ice rheology, the notion of solutions to the original model is unclear. Instead, an approximating system, which captures current numerical study, is proposed. The well-posedness theory of such a system provides a first-step groundwork in both numerical study and future analytical study.


The sea-ice dynamic-thermodynamic model
Global climate changes, especially global warming, have large impact on the Arctic sea-ice, which has, in return, determining effects on not only global climate but also the local and global ecosystem, human activities etc. (see e.g., [26]). If the problem is statically determinate, as pointed out in [23], a sea-ice dynamical model based on the viscous-plastic rheology was introduced in [13], where the thickness of ice plays an essential role in the thermodynamics, and characterizes the strength of the ice interaction (i.e., ice rheology). The velocity of sea-ice u is described by two-dimensional momentum balance equations, where the viscosity effect is characterized by a viscous-plastic rheology, and the strength of viscosity depends on the thickness of ice. The mean ice thickness h and the compactness of ice A are described by two continuity equations with thermodynamic source terms. That is, with a simplified ice rheology (see (1.11a), below), the above quantities are governed by the coupled system, m(∂ t u + u · ∇u) + ∇p = div S + F, (1.1a) respectively. In addition, v ⊥ = (−v 2 , v 1 ) ⊤ for any vector v = (v 1 , v 2 ) ⊤ ; ρ ice , ρ a , ρ w represent the density of ice, air, and water, respectively; c p , c a , C a , C w are the thermodynamic constants; and U g , U w , φ, θ denote the velocity and stress angle of the air and the water, which, for simplicity of presentation, are assumed to be constant in this paper. System (1.1) is used to study simulation the evolution of sea-ice in numerical study. For instance, the model successfully reproduces many of the observed features of the circulation and thickness of the Arctic ice cover in [13]. Hibler's model of sea-ice dynamics is the foundation for further model developments, including the elastic-viscous-plastic sea-ice dynamics model as in [15], the Maxwell elasto-brittle rheology model as in [4], and models with leads, ridging or tensile failure as in [24,28]. See [21] for a summary of classical models with different descriptions of ice distribution, and [1,3,12,14,18,20,22,27] and the references therein for further model development and computational investigation.
Despite the steady advances that have been made in the modeling and simulation of sea ice, mathematical analysis is much less developed in this field. It is the main objective of this paper to provide rigorous mathematical analysis of Hibler's model as a first step in this direction and as a basis for further investigation of next-generation sea-ice rheologies.
In particular, the fundamental problem of well-posedness of solutions to system (1.1) is widely open, which is related to the validity of the model as pointed out by [23]. In [9,10], the authors study the loss of hyperbolicity of the linearized system of the original Hibler's model around divergent flows, and show that the system is ill-posed. This work is further discussed in [5,11,16,25] from various perspectives. These studies do not contradict the local well-posedness result established in this paper, for the following reasons: • Instead of the original Hibler's model, we consider a regularization of Hibler's model, which, as shown below in the paper, preserves the parabolicity of the momentum equations in a Sobolev space with enough regularity. This is very different from the hyperbolic equations considered in the ill-posedness studies; • Instead of linear analysis, we consider the nonlinear well-posedness theory for regularized Hibler's model in the Hadamard sense, including the existence, the uniqueness, and the continuous dependency on initial data in the suitable Sobolev space as shown in Theorem 1.1.
We would like to point out that the main challenge in establishing the well-posedness theory is the singularity arising in the stress tensor (1.2c) when |∇u| → 0 + . In fact, among the numerical investigations, such singularity is usually truncated, i.e. regularized, by replacing it with its strictly positive approximation (e.g., max{|∇u|, ∆ min } or |∇u| 2 + ε 2 ).
Notably, we would like to point out an investigation of very singular diffusion equations in [7,8], where the authors discuss the notion of solutions to Similarly, the positive one-homogeneity of the potential related to (1.2c) calls for a subdifferential formulation of the problem, however set in the Eulerian frame. We leave such investigation to our future study. In this paper, due to the obstacles mentioned above, we propose to study the following regularized approximating problem of (1.1): for ε, ω ∈ (0, 1), where m, p, and F are as in (1.2a), (1.2b), and (1.2d), respectively, and To be more precise, we will establish the local in time well-posedness of strong solutions to (1.3) in domain Ω := T 2 ⊂ R 2 : In addition, we assume that for some constants f , f ∈ R, M f ∈ (0, ∞). Then there exists a unique strong solution (u, h, A) to system (1.3) in [0, T ] × Ω, for some T ∈ (0, ∞) depending on initial data, with where C in ∈ (0, ∞) is some positive constant depending only on initial data. Moreover, the solution is stable with respect to perturbation of initial data.
Now, let us explain our strategy. Instead of directly constructing solutions to system (1.3), we consider another regularized system, parametrized by (µ, λ, ι, ν) ∈ (0, 1) 4 : where m, p, F, and χ ω A are as in (1.3) and (1.4), and (1.11e) We will construct solutions to system (1.10) through a contraction mapping argument. That is, we consider a "linearization" of (1.10), and establish a contraction mapping with respect to L 2 topology with bounds in a smooth function space. Then with a uniform-in-(µ, λ, ι, ν) estimate, we will be able to pass the limit (µ, λ, ι, ν) → (0 + , 0 + , 0 + , 0 + ), and eventually construct the strong solution to (1.3). The proof of Theorem 1.1 is then finished by showing the uniqueness and continuous dependency on the initial data. We would like to mention that the key ingredient in establishing the wellposedness of solutions involves showing the monotonicity of S ε (·) in ∇u, which is not trivially obvious due to the fact that S ε (·) is nonlinear in ∇u.
In particular, we will require the inequality of the type We successfully establish this inequality by writing S ε (p 1 , ∇u 1 )−S ε (p 2 , ∇u 2 ) in a symmetric form (see (4.29), below). We would like to make some remarks before going into details of the proof. Our ice rheology (1.4a) is a simplified version of the one from [13]. For some technical reasons, we are not sure whether Theorem 1.1 will apply to the original ice rheology from [13]. We have not successfully established a proper uniform-in-ε estimates of the solutions to (1.3). Therefore, we have not yet been able to establish a proper notion of solutions to the original system (1.1). However, our approximation (1.3) agrees with the most common numerical approaches to (1.1), which, as we explain before, is restricted to a truncated ice rheology. Thus, in this sense, our analytical results provide a solid ground for current numerical schemes of (1.1). Another issue is that we only consider the case when h in ≥ h > 0, i.e., there is no absence of ice in the domain of study. To carry out the limit h → 0 + , more comprehensive a priori estimates are required. We leave this to future study.
Recently we have learnt an independent study [2] by Brandt, Disser, Haller-Dintelmann, and Hieber about similar model, where the domain boundary and boundary conditions are taken into account instead of periodic domain. It is worth pointing out that in our regularized system (1.3) the governing equations for the evolution of h and A remains hyperbolic, and therefore system (1.3) is a mixed type system, while the regularized system in [2] is parabolic in all its components. In particular, due to the hyperbolicity, system (1.3) is expected to have a completely different long-time dynamics than those investigated in [2]. Moreover, the additional dissipation introduced in [2] allows the authors to show global existence for small initial data. This paper is organized as follows. In the next subsection, we will summarize some notations used in this paper. In Section 2, we will detail the approximation scheme to (1.10). In Section 3, we establish the well-posedness of solutions to (1.10) via a contraction mapping argument. Finally in Section 4, we establish the uniform-in-(µ, λ, ι, ν) estimates, and pass to the limit (µ, λ, ι, ν) → (0 + , 0 + , 0 + , 0 + ) to show the existence of solutions to (1.3). The well-posedness of solutions is then established in Section 4.3
2 An approximation scheme to solve (1.10) 2.1 A "linearization" of (1.10) Given u o , assumed to be smooth enough, we consider first the following coupled hyperbolic system where S hm,ω,ν , S Am,ω,ν , χ ω hm , and χ ω Am are defined as in (1.11b), (1.11c), (1.11e), and (1.4d), with h and A replaced by h m and A m , respectively. Here we use the subscript m (short for 'mapping') and the superscript o (short for 'origin') to label outputs and inputs in our contraction mapping.
We claim that, at least locally in time, there exists a unique solution (h m , A m ) to (2.1a) and (2.1b) with proper initial data, for smooth enough u o . (h m , A m ) can be arbitrarily regular, provided that u o and initial data are regular enough. We leave the investigation of the regularity of (h m , A m ) in the subsequent sections.
We remark that such claims follow from the standard well-posedness theory of hyperbolic equations (see, e.g., [17]). Hence the proof is omitted.
Let (h m , A m ) be the solution to (2.1a) and (2.1b) as above, and consider the following equation: where p m , S ε,µ,λ,m , and F m are defined as in (1.2b), (1.11a), and (1.2d), with h, A, and u replaced by h m , A m , and u o , respectively.
To solve the linear equation (2.1c) by, e.g., a Galerkin method, one will need to deal with the possible degeneracy of h m . For this, we subsequently show that for u o smooth enough, with appropriate initial data, h m and A m satisfy certain non-degeneracy property.

Non-negativity and uniform bound of
In this subsection, we show that 0 ≤ A m ≤ 1 for a smooth enough u o . In fact, we only require that for some T > 0.

Non-negativity of A m :
Taking the L 2 -inner product of (2.1b) with −A − m leads to, after applying integration by parts in the resultant Non-negativity of 1 − A m : Consider the following equation for 1 − A m , derived from (2.1b): As before, taking the L 2 -inner product of (2.4) with −(1 − A m ) − , after applying integration by parts, leads to Then as before, after applying Grönwall's inequality to (2.5), one can conclude A m ≤ 1.

Non-negativity, lower and upper bounds of h m
Let h, h ∈ [0, ∞) be the lower and upper bounds of h in , respectively, i.e., (1.6)). In this section, we will show that locally in time. Again we assume that u o has the regularity (2.2).

Non-negativity of h m :
After applying the L 2 -inner product of (2.1a) with −h − m and applying integration by parts in the resultant, one has since the term S hm,ω,ν (−h − m ) vanishes. Therefore, applying Grönwall's inequality to (2.6), as before in (2.3), eventually implies h m ≥ 0.

Lower and upper bounds of h m :
Since Thus, integrating in the above inequation along the characteristic path given by u o yields Similarly, one can show that Then it immediately follows that 1 4 h ≤ h m ≤ 4h provided that the following conditions are satisfied: (2.9)

Non-vanishing total ice mass
Therefore, 2.5 Well-posedness of (2.1c) with strictly positive ice mass 3 Well-posedness of solutions to (1.10) with h > 0 and ι > 0 fixed In this section, we aim at showing that the map defined by where u m is the unique solution to (2.1c) with h m and A m being solutions to (2.1a) and (2.1b), respectively, is bounded in X T * and contracting with contraction for some T * ∈ (0, ∞) to be determined. Throughout this section, unless stated otherwise, the initial data for u m , u o , h, and A are assumed to be u in , u in , h in , and A in , given in Theorem 1.1, respectively. Consequently, one can apply the Banach fixed-point theorem, i.e., the contraction mapping theorem, to show the existence of solutions to system (1.10).
Let c in ∈ (0, ∞) be the bound of the initial data defined by

Uniform bounds
Estimates for h m and A m Aside from the point-wise estimates deduced in Sections 2.2 and 2.3, we shall need a uniform H 1 -estimate for A m and h m . We record the equation after applying ∂ ∈ {∂ x , ∂ y } to (2.1a), as follows: Then taking the L 2 -inner product of (3.5) with 4 ∂h m 2 ∂h m leads to, after applying integration by parts, (3.6) Meanwhile, simple calculation shows that where we have used (1.7). Consequently, one concludes from (3.6) that (3.7) The estimate for ∇A m is obtained from (2.1b) in a similar fashion, we record it here: where On the other hand, in direct consequence of equations (2.1a) and (2.1b), one has where we have used the fact that 0 ≤ A m ≤ 1 and (1.7).
Estimates for u m Taking the L 2 -inner product of (2.1c) with 2u m + 2∂ t u m − 2∆u m leads to, after applying integration by parts, (3.13) We obtain the following estimates for the R j terms by applying Hölder's inequality and the Sobolev embedding inequality: To deduce the above estimates, consider h > 0 and let t satisfy (2.9) and (2.11). Therefore, the estimates in Section 2.3 guarantee that 0 < 1/4h ≤ h m ≤ 4h < ∞. Consequently, (3.13) yields, after applying the Sobolve embedding inequality and Hölder's inequality, (3.14) Furthermore, consider t small enough such that where we have applied Hölder's inequality. Then (3.9) and (3.12) imply that, after applying the Sobolev embedding inequality, Consequently, (3.14) yields the following estimate: provided that t is small enough and where we have made the choice where the right-hand side is as in (3.17). Then (2.9), (2.11), (3.15), and (3.17) imply that, there exists T * ∈ (0, ∞) such that for t ∈ [0, T * ]. In addition, using equation (1.10), it is easy to obtain Therefore, M, defined in (3.1), maps X T * into itself for such choices of T * and c o . We remark here that, c 0 → ∞ as ι → 0 + , i.e., the estimates we obtain here depend on ι > 0. We will remove the dependency of ι in Section 4.

Contraction mapping and well-posedness
For j = 1, 2, consider u o j ∈ X T * satisfying (3.4), and let h m,j , A m,j , and u m,j = M(u o j ), be the solutions to (2.1a), (2.1b), and (2.1c), respectively, with u o replaced by u o j and with the same initial data. Then we have the estimates of h m,j , A m,j , and u m,j as in Sections 2.2 and 2.3, as well as (3.16) and (3.19a).
In the following, let σ ∈ (0, 1) be a constant to be determined later.   (3.23) In the following, we sketch the estimates of the R j terms by applying Hölder's inequality and the Sobolev embedding inequality: where we have used the identity where we have also employed Young's inequality. Taking the L 2 -inner product of (3.22c) with 2δu m and applying integra-tion by parts in the resultant yields (3.27) In the following, again, we sketch the estimates for the terms R j by applying Hölder's inequality, the Gagliardo-Nirenberg inequality, and the Sobolev embedding inequality: To estimate R 22 , we rewrite it as Therefore, applying Hölder's inequality and the Sobolev embedding inequality implies where we have used the identity δ g for g = ∇u o + (∇u o ) ⊤ and div u o I 2 , respectively. Then, after substituting the bounds in (3.16) and (3.19a) and applying interpolation inequalities, one can obtain from (3.27) that where Young's inequality is applied. Thus, after substituting (3.26) into (3.28) and applying Grönwall's inequality to the resultant, one has sup 0≤s≤t δu m (s) 2 Therefore, after choosing σ and t small enough, one can conclude that (3.29) Now we update the smallness of T * , so that (3.29) holds true for t ∈ (0, T * ]. Then the map M, defined in (3.1), is contracting with constant 1/2. By means of Banach's fixed point theorem, we conclude that there exists a unique solution to (1.10) in X T * . What is left is to show that such solutions are stable. Namely, they continuously depend on the initial data. Let (u j , h j , A j ) be two solutions to (1.10), associated with initial data (u in,j , h in,j , A in,j ), j = 1, 2, satisfying (3.30) Hence, we have established the local-in-time well-posedness of strong solutions to system (1.10). We would like to remind readers that the estimates obtained in this section depend on (µ, λ, ι, ν). In the next section, we aim at removing such dependency. We shall only present the uniform-in-(µ, λ, ι, ν) a priori estimate in this subsection, based on which the standard different quotient argument can be established.
Throughout this section, we use the notation X Y to represent X ≤ CY for some generic constant C ∈ (0, ∞), which may be different from line to line, and depend on ε, ω, h, h, but is independent of (µ, λ, ι, ν).
To begin with, let and (4.2) One can easily check that E and E are essentially equivalent in the sense that estimates on one imply estimates on the other. Indeed, it is trivial that E E. On the other hand, applying integration by parts yields that (4.3) Therefore, we have (4.4)

Estimates for h and A
It is easy to check that (2.3), (2.5), (2.6), (2.7), and (2.8) also hold true with A m , h m , u o replaced by A, h, u, respectively. Therefore, for s ∈ (0, t) with t satisfying (2.9), with u o replaced by u, we have Notice that the smallness of t here is independent of (µ, λ, ι, ν). Next, we shall establish the regularity estimates of A and h. Indeed, after applying ∂ 3 to (1.10b) and (1.10c), one can obtain the following equations: (4.6b) Then, applying the L 2 -inner product of (4.6a) and (4.6b) with 2∂ 3 h and ∂ 3 A, respectively, and integration by parts in the resultant leads to (4.7b) Directly applying Hölder's inequality and the Sobolev embedding inequality leads to the following estimates: (4.8) Similarly, Therefore, after substituting estimates (4.8) and (4.9) into (4.7a) and (4.7b), one can derive that Similar estimates also hold for lower order derivatives. Hence we have shown that for some constant C ω,h,h ∈ (0, ∞), independent of ι and ν. Consequently, applying Grönwall's inequality concludes that (4.10) Estimates for u After applying ∂ 3 to (1.10a), one can obtain the following equation: Then, applying the L 2 -inner product of (4.11) with 2∂ 3 u and integration by parts in the resultant leads to (4.12) The estimates of I j , j ∈ {8, 11, 12}, are standard, which we will record below. Applying Hölder's inequality and the Sobolev embedding inequality yields that (4.13) To estimate I 13 , notice that where l.o.t represents lower order terms of u. Direct calculation yields that which implies Therefore, we have . (4.14) In order to estimate I 10 , we first rewrite I 10 as follows, where we have applied integration by parts. Next, we will use equation (1.10a) to substitute ∂ t u and ∂ t ∂u in (4.15). Indeed, after rearranging (1.10a), it follows Then similarly as before, directly applying Hölder's inequality and the Sobolev embedding inequality leads to, Therefore, one can derive that, (4.16) Lastly, we will estimate I 9 . Notice that, Denote by Du ∈ {∇u+∇u ⊤ , div u}. In this notation, estimating I 9 amounts to determining an estimate for Direct calculation shows that To sum up, after substituting estimates (4.13), (4.14), (4.16), (4.17),and (4.18) into (4.12), and applying Young's inequality, one can derive that which implies, recalling ι ∈ (0, 1), for some constant C ε,h,h ∈ (0, ∞), independent of µ, λ, ι, and ν.
Similar estimates also hold for lower order derivatives. Thus one can conclude that, for ι ≪ 1 small enough, sup 0≤s≤t u(s) 2

Uniform estimates
The summation of (4.10) and (4.19) leads to where we have applied (4.3) and Young's inequality in the second inequality. Consequently, for t small enough, independent of µ, λ, ι, ν, one can conclude that 20) and, thanks to (4.3), for some constant C in ∈ (0, ∞), depending only on ε, ω, h, h, and Thus we have established the (µ, λ, ι, ν)-independent estimates. Therefore, together with the well-posedness theory in Section 3 and continuity arguments, the existence time of solutions to (1.10) can be extended to some T * * ∈ (0, ∞), independent of (µ, λ, ι, ν), which might be larger than T * .

Well-posedness of solutions for system (1.3)
To deduce the well-posedness of solutions to system (1.3), it remains to establish the uniqueness and the continuous dependency of solutions on initial data. Indeed, this can be done following similar arguments as in Section 3.2, which we will sketch below.