Existence of Solutions to a Phase-Field Model of 3D Grain Boundary Motion Governed by a Regularized 1-Harmonic Type Flow

In this paper, we propose a quaternion formulation for the orientation variable in the three-dimensional Kobayashi–Warren model for the dynamics of polycrystals. We obtain existence of solutions to the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-gradient descent flow of the constrained energy functional via several approximating problems. In particular, we use a Ginzburg–Landau-type approach and some extra regularizations. Existence of solutions to the approximating problems is shown by the use of nonlinear semigroups. Coupled with good a priori estimates, this leads to successive passages to the limit up to finally showing existence of solutions to the proposed model. Moreover, we also obtain an invariance principle for the orientation variable.


Introduction
In [18] and [25,26] two very similar models for the dynamics of polycrystals in the three dimensional space were introduced.The authors generalized the existing two dimensional model by Kobayashi et al [19,20] to the case of 3D-crystals.In essence, the 2D-model is the L 2 -gradient descent flow of the following energy functional: In the context, the unknowns η = η(t, x) and θ = θ(t, x) represent, respectively, the "orientation order" and the "orientation angle" in a polycrystal.η = 1 corresponds to a completely ordered state while η = 0 corresponds to the state where no meaningful value of mean orientation exists.In the original work [19], G(s) = 1 2 (1 − s) 2 ensures that the ordered state η = 1 is stable.
In order to generalize the model, one needs to consider orientations in 3D and misorientations, since the term |∇θ| represents the misorientation in a short scale.In 3D, orientations are elements of SO(3), the special orthogonal group in R 3 .In [18], the term |∇θ| is substituted by the corresponding Euclidean norm in R 9 ; i.e. ||∇P || R 9 := 3 i,j=1 |∇p i,j | 2 1 2 for P = [P ] i,j ∈ SO (3).Then, one has to compute the gradient descent flow for the constrained energy, thus ensuring that the solutions for the orientation variable still belong to SO (3).
In [25,26], instead, a quaternion representation is used for SO (3).Then, by the use of the fact that quaternions can be identified as elements in the unit sphere in R 4 , i.e. S 3 , the authors replaced the term |∇θ| by the Euclidean norm of the gradient of the quaternion: i.e. |∇q| := 3 i=0 |∇q i | 2 1 2 , for q = (q 0 , q 1 , q 2 , q 3 ) ∈ S 3 .We take the point of view of [25,26] and we consider the following energy functional; constrained to functions with values in the unit sphere of R M with 1 < M ∈ N, i.e. S M −1 : if η ∈ H 1 (Ω) and u ∈ H 1 (Ω; S M −1 ), +∞, otherwise; ( where H 1 (Ω; S M −1 ) := ũ ∈ H 1 (Ω; R M ) |ũ| = 1 a.e. in Ω .Now, on any time interval (0, T ) with a finite constant T > 0, the corresponding L 2 -gradient descent flow of F can be formally written as the following system of PDEs: We point out that the writing is purely formal since there are some undefined terms such as ∇u |∇u| .The precise meaning of a solution to the system is given in Section 3.
A natural way to study this type of restricted functionals in the sphere is to relax the constraint via a Ginzburg-Landau approximation (see [5]); i.e. instead of considering u ∈ H 1 (Ω; S M −1 ), one lets u ∈ H 1 (Ω; R M ) and adds the following term to F : After obtaining well posedness to the gradient descent flow, then the strategy is to let δ → 0 + and to show convergence of a subsequence to the corresponding solution to the system (P).We follow exactly this strategy, but, due to the non-differentiability of the Euclidean norm at the origin, we also need to approximate it by a sufficient smooth term; i.e. we replace Due to technical reasons for a possible future study, we need to perform an extra approximation.One might study the limit problem when κ → 0 + in (1.1); i.e. the case in which u ∈ BV (Ω; SO(3)).In this case, the term in the energy corresponding to Ω α(η)|∇u| needs to be replaced by its relaxed functional.This relaxed functional has a jump part that strongly depends on the metric considered in SO(3) (see [14]).By the considerations stated in Appendix B, and in order to uniquely identify a rotation as an element in S 3 , we need to restrict the solutions in the quaternion representation to lie in the open upper hemisphere S 3 + := {p = (p 1 , p 2 , p 3 , p 4 ) ∈ S 3 : p 1 > 0}.For the sake of generality, we will consider the more general setting u ∈ S M −1 instead of S 3 .In Appendix B, we give a maximum principle which ensures that, if the initial datum is in a certain compact subset of S M −1 + , then the solution also does.For the proof of this result, we need a technical restriction; namely, continuity of the solutions.Therefore, we need to perform an extra approximation to F , by adding the following term to the energy functional: 1 We stress that this extra regularization is only a technical tool to prove the maximum principle in Theorem 6 and it is not needed for any of the rest of the results in the present manuscript.
The plan of the paper is the following one: First of all, in Section 2, we prescribe some notations, and recall some results about multi-vectors that are used in the paper.In Section 3, we set up our main assumptions, and we state the Main Theorem, as the principal result of this paper.In Section 4 we consider the complete energy functional (1.1), which we call the free energy, we define our notion of solution to its L 2 -gradient descent flow; i.e to the system named by (P) κ ε,ν,δ .Then, we prove with the help of an auxiliary convex energy functional, that the system (P) κ ε,ν,δ admits a unique solution for sufficiently smooth initial data (see Theorem 2).Moreover, stability with respect to the parameter ν is obtained.
Section 5 is devoted to the proof of Main Theorem, i.e. the proof of existence of solution to the L 2 -gradient descent flow of F .First of all, an energy inequality together with the corresponding uniform estimates (in the parameters ε and δ) are obtained for solutions to (P) κ ε,ν,δ .They lead to convergence, first with δ → 0 + and up to subsequences, to a solution to the system (P) ε,ν ; i.e, to the gradient descent flow of the restricted energy functional: Moreover, solutions are shown to be continuous.Therefore, the maximum principle stated and proved in the Appendix applies and we can move further by letting ν → 0 + .Then, we obtain existence of solutions to the system (P) ε ; i.e. to the gradient descent flow of F ε := F ε,0 .The final step is to let ε → 0 + , thus obtaining existence of solutions to the gradient descent flow of F ; i.e. to (P).We point out that, since the successive convergences for the orientation variable (with ν → 0 + and ε → 0 + ) also hold a.e. in space time, then, the final solutions also satisfy the maximum principle.Finally, we added two Appendices.In the first one, we recall the concept of Mosco convergence and some results related to it that we use in the paper.The second one is devoted to the discussion about the relationship between rotations in SO(3) and their representation as quaternions.It is there where we prove our maximum principle.

Abstract notations
For an abstract Banach space X, we denote by • X the norm of X, and by •, • X the duality pairing between X and its dual X ′ .In particular, when X is a Hilbert space, we denote by (•, •) X the inner product of X.Moreover, when there is no possibility of confusion, we uniformly denote by |•| the norm of Euclidean spaces, and for any dimension d ∈ N, we write the inner product (scalar product) of R d , as follows: For any subset A of a Banach space X, let χ A : X −→ {0, 1} be the characteristic function of A, i.e.: . In particular, when all X 1 , . . ., X d coincide with a Banach space Y , we write: For a proper, lower semi-continuous (l.s.c.), and convex function Ψ : X → (−∞, ∞] on a Hilbert space X, we denote by D(Ψ) the effective domain of Ψ.Also, we denote by ∂Ψ the subdifferential of Ψ.The subdifferential ∂Ψ corresponds to a weak differential of convex function Ψ, and it is known as a maximal monotone graph in the product space X × X.The set D(∂Ψ) := z ∈ X | ∂Ψ(z) = ∅ is called the domain of ∂Ψ.We often use the notation "[w 0 , w * 0 ] ∈ ∂Ψ in X × X ", to mean that "w * 0 ∈ ∂Ψ(w 0 ) in X for w 0 ∈ D(∂Ψ)", by identifying the operator ∂Ψ with its graph in X × X.
Next, for Hilbert spaces X 1 , • • • , X d , with 1 < d ∈ N, let us consider a proper, l.s.c., and convex function on the product space Besides, for any i ∈ {1, . . ., d}, we denote by ∂ w i Ψ : As is easily checked, where It should be noted that the converse inclusion of (2.1) is not true, in general.

Multi-vectors
Here we recall some definitions and basic properties about multi-vectors that we need in our analysis.We refer to e.g.[12, Chapter 1] and [9, Chapter 1] for details.
Let m ∈ N. The spaces Λ 0 (R m ) and Λ 1 (R m ) are defined as: For any integer 2 ≤ k ≤ m, the k-th exterior power of R m , denoted by Λ k (R m ), is defined as a set spanned by elements of the form: called "generators", which are subject to the following rules: for any basis e 1 , . . ., e m of R m , the set forms the basis of Λ k (R m ), where 2) The elements of Λ k (R m ) are called multi-vectors (or k-vectors), and Λ k (R m ) is a vector space of dimension m k .Given k, ℓ ∈ {0, . . ., m} with k + ℓ ≤ m, there exists a unique bilinear map (λ, µ) (2. 3) The Hodge-star operator: ), which is defined on the basis as: for all permutations {α 1 , . . ., α m } of {1, . . ., m}, having positive signature. (2.4) In particular, in what follows we will systematically identify Λ m−1 (R m ) with R m and Λ m (R m ) with R. We will use the following well known formulas: (see e.g.[9, (1.64)]) and Introducing the inner product on generators

Vector valued functions
Let X be a Banach space with dual X ′ and let U ⊂ R d be a bounded open set endowed with the Lebesgue measure L d .A function u : U → X is called simple if there exist x 1 , . . ., x n ∈ X and U 1 , . . ., U n L m -measurable subsets of U such that u = n i=1 x i χ U i .The function u is called strongly measurable if there exists a sequence of simple functions {u n } such that u n (x)−u(x) X → 0 as n → +∞ for almost all x ∈ U.If 1 ≤ p < ∞, then L p (U; X) stands for the space of (equivalence classes of) strongly measurable functions u : U → X with Endowed with this norm, L p (U; X) is is a Banach space.For p = ∞, the symbol L ∞ (U; X) stands for the space of (equivalence classes of) strongly measurable functions u : If U = (0, T ) with 0 < T ≤ ∞, we write L p (0, T ; X) = L p ((0, T ); X).
) is isometric to a subspace of (L p (0, T ; X)) ′ , with equality if and only if X ′ has the Radon-Nikodým property (see for instance [10]).
We consider the vector space D(U; X) := C ∞ 0 (U; X), endowed with the topology for which a sequence ϕ n → 0 as n → +∞ if there exists K ⊂ U compact such that supp(ϕ n ) ⊂ K for any n ∈ N and D α ϕ n → 0 uniformly on K as n → +∞ for all multi-index α.We denote by D ′ (U; X) the space of distributions on U with values in X; that is, the set of all linear continuous maps T : D(U; X) → R. As is well known, L p (U; X) ⊂ D ′ (U; X) through the standard continuous injection.Given T ∈ D ′ (U; X), the distributional derivative of T is defined by for any ϕ ∈ D(U; X) and any i ∈ {1, . . ., d}. (2.13) .

Main Theorem
We start with setting up the assumptions in the principal part of this paper.The assumptions also fix the notations in the system (P), and in its approximating problems.Ω T := (0, T ) × Ω, and Γ T := (0, T ) × Γ, R) are fixed functions, such that: • α ′ (0) = 0, α ′′ ≧ 0 on R, and α and αα ′ are Lipschitz continuous on R.
) is a continuous convex function, defined as: We let ̟ δ ∈ C 1 (R M ; R M ) be the gradient of Π δ , i.e.: ), and if ε = 0, then the corresponding function f 0 coincides with the (Euclidean) norm Additionally, concerning the subdifferentials ∂f ε , ε ≥ 0, we can observe that: where Sgn M,N : R Next, we define the notion of solution to our system (P).
is called a solution to the system (P), iff.: and there exist functions B * ∈ L ∞ (Ω T ; R M N ) and µ * ∈ L 1 (0, T ; L 1 (Ω)), such that: and Now, the Main Theorem of this paper is stated as follows.

Approximating problem
In this Section, we consider the approximating problem to our system.Let us assume (A0)-(A6), and fix constants ε ≥ 0, δ ≥ 0, and ν ≥ 0. On this basis, the approximating problem consists in the following system of parabolic PDEs, denoted by (P) κ ε,ν,δ : The approximating problem (P) κ ε,ν,δ is derived as a gradient descent flow of a free energy, which is defined as: for any ϕ ∈ V , a.e.t ∈ (0, T ), subject to η(0) = η 0 in H; and there exists B * ∈ L ∞ (Ω T ; R M N ), such that: Additionally, we note that our system (P) κ ε,ν,δ can be reformulated as the following Cauchy problem of evolution equations in the Hilbert space X, denoted by (CP) κ ε,ν,δ .

Proofs of Theorems 1 and 2
For the proofs of Theorems 1 and 2, we prepare some Lemmas and Remarks.
Proof.As is easily checked, n=1 of continuous convex functions: converges to the continuous convex function: converges to the convex function: converges to the convex function: 1 on R M , in the sense of Mosco, as n → ∞.
In the meantime, from (A4), one can see that: ≤ 1, a.e. in Ω, for any n ∈ N, ♯ 5) the sequence of convex functions: converges to the convex function: on L 2 (Ω; R M N ), in the sense of Mosco, as n → ∞.
Proof of Lemma 4 (I).We set: and prove this R 0 is the required constant.Let us assume: Then, with ) N by (A3) and Hölder's inequality, and A3), (A4) and Hölder's inequality.Due to (4.25), using Young's inequality, the inequalities in (4.26) lead to: which implies the (strict) monotonicity of the operator Proof of Lemma 4 (II).Let R 0 > 0 be the constant defined in (4.25).Then, in the light of the inclusion Also, invoking [3, Theorem 2.10] and [6, Corollary 2.11], we also have: Then, item (II) is a direct consequence of the above ♯ 6) and ♯ 7).
Based on the above Lemmas, Theorems 2 and 1 are proved as follows.

Proof of Main Theorem.
In this Section, we show the proof of Main Theorem.To see this, we take the limit in (P) κ ε,ν,δ as δ, ν, ε → 0 + , respectively.At first, we derive an energy inequality for F κ ε,ν,δ and a priori estimates for the approximate solutions.
In this subsection we study the limit problem, as δ → 0, in (P) κ ε,ν,δ , assuming 0 < ε < ε 0 ; i.e. we solve the following problem.Note that the dependence on κ is not anymore needed or used.Therefore, we remove it.
together with: and ) where the constant C > 0 is independent of ε.
Proof.By the same argument as that in the proof of Theorem 3, there exists a subsequence {U ε,νn } n and a function as ν → 0.Then, we also see that After that, the proof follows the same way as Theorem 3 using Lemma 8 instead of Lemma 7. In fact, we have for ω ∈ L ∞ (0, T ; H 1 (Ω; Λ 2 (R M ))), and The rest of the proof is the same as that of Theorem 3 and we omit the details.Finally, let u 0 ∈ S M −1 +,r for r ∈ (0, 1).Therefore, Theorem 6 (maximum principle) implies that u ε,ν ∈ S M −1 +,r a.e. in Ω T , which, by (5.22) implies that u ε ∈ S M −1 +,r a.e. in Ω T .
Finally, we solve the initial system (P) by letting ε → 0. As ε → 0, the limit problem is formulated as follows.
Next, we set f ε := µ ε u ε .Then, f ε is uniformly bounded in L 1 (0, T ; L 1 (Ω; R M )) for ε.By Lemma 9, it follows that ∇u εn → ∇u strongly in L 1 (0, T ; L 1 (Ω; R M N )) as n → ∞.The rest of the proof is the same as that of Theorem 3 and we omit it.Finally, in the case that u 0 ∈ S M −1 +,r a.e. in Ω T , by Theorem 5 and (5.25) we easily get the conclusion.
u 0 ∈ S 3 + and with the geometry of S 3 + , which is much simpler and it has been used much wider in variational constraint problems than that of SO(3).However, in order that the solution to the flow still belongs to S 3 + and therefore it can be identified with a rotation, we need the following result: Theorem 6. (Maximum principle) Suppose that u 0 ∈ B g (p 0 ; R), with R < π 2 (equivalently w P 0 ∈ [0, r] with r < π).Then, the solution to (P) ε,ν satisfies u ε,ν ∈ B g (p 0 ; R) , a.e. in Ω.
We choose on B g (p 0 ; π 2 ) a polar coordinate system p → (p r ; p θ 1 , . . ., p θ M −2 ) centered at p 0 .Next, we compute the second equation in (P) ε,ν for the radial coordinate: The metric in the polar coordinates around the north pole is the following one: .
a bounded domain with Lipschitz boundary Γ := ∂Ω and unit outer normal n Γ .Besides, let: