EVOLUTION OF SPIRAL-SHAPED POLYGONAL CURVE BY CRYSTALLINE CURVATURE FLOW WITH A PINNED TIP

. Evolution of convex polygonal spiral with ﬁxed center by crystalline eikonal-curvature ﬂow is considered. In this evolution we consider a new facet of the polygonal curve generates from center when a facet associated with the center evolves with enough length, which is equal to the length of the facet in Wulﬀ shape of energy density function. We prove the existence, uniqueness and intersection free of solution to our formulation globally-in-time. In the proof of the existence we also prove that new facets are generated repeatedly in time. The important property for intersection-free result is monotonicity property such that the normal velocity of every facets are positive after the next new facet is generated, so that the center is always behind of the moving facets.

1. Introduction. Burton, Cabrera and Frank [1] proposed a theory of crystal growth with aid of screw dislocations in 1951. According to the theory, a monomolecular step is provided by a screw dislocation across with the crystal surface. Atoms on the surface are caught by kinks, which is a corner of atoms in the step, with a heigher probability when they are close to the steps, and then results in an evolution of steps. The dynamics of steps in this setting is given as [1], where V and H, respectively, are the normal velocity and the curvature of the curve drawn by the steps, and U is a constant denoting the driving force of the evolution. Note that the directions of V and H are inverse, so that the above equation should be a parabolic type equation. There is a nice review paper [2] on its mathematical modelling as well as computational methods. One often can find a polygonal spiral steps on the growing crystal surface: see e.g. [21]. It is caused by the anisotropy of the surface energy by the geometry of the structure of atoms. Anisotropic surface energy of a curve S should be given by (2) E γ (S) = S γ(n)dσ by a density function γ : S 1 → (0, ∞), where dσ is the line element. Then, we obtain the weighted curvature H γ of curve S with surface energy density γ as the first variation of E γ , i.e., H γ (n, ∇n) = δE γ δS (S).
Moreover, we consider the mobility of the evolution reflects the anisotropy of the lattice of atoms as the coefficient in front of the normal velocity. Then, we obtain the generalized evolution equation by anisotropic curvature and velocity. The equation (1) is regarded as the above equation with γ ≡ 1 and β ≡ 1. When the polygonal curve appears provided that the curve S evolves by (3), then its stationary solution, called "Wulff shape", should be a polygon, where x · y denotes the usual inner product for x, y ∈ R 2 . However, it is well-known that γ is not only not smooth, but also possibly not convex even if W γ is a convex polygon. At least it is also well-known that piecewise linear γ implies convex polygonal W γ . We call the energy (2) having convex polygonal W γ as crystalline surface energy, and the motion of polygonal curves by crystalline surface energy as crystalline motion. In this paper, we are interested in evolution of spirals with crystalline surface energy. If γ is smooth and convex, then several PDE approachs formulating interface evolution equation with (3) are proposed. There is a nice review book [5] for the theory of interface evolution equation with PDE approach. However, because of the non-smoothness or non-convexity of γ, a lot of PDE approach tracking the evolution of curves does not work well for the crystalline motion. Taylor [22] proposed an ODE approach tracking the evolution of polygonal curve with (3) by a system of ordinary differential equation on the length of each facet. In the theory by [22] we introduce the crystalline curvature as the ratio of the evolving facets and that in W γ with the same orientation. Then, we calculate the extending speed of the facets from the displacement of the evolving facets by (3). Then, we describe the evolution of curves by the length and tangential direction of facets of the evolving curves. There is a good book [8] for details of the ODE approach to the crystalline motion by interfacial curves. Mathematical analysis for the crystalline motion of interfacial curves has been done well. Ushijima, Yagishita, Yazaki and the first author [14] proved the existence of non-convex self-similar solution to the crystalline curvature flow, which violates the convexity phenomena as in the isotropic curvature flow by [7]. Then, the first author [10] classified the motion of closed polygonal curves by the crystalline curvature flow. For the crystalline eikonal-curvature flow, the first author [12,11] investigated the behavior of V-shaped solution. On one hand there is a few works on level set approach [4,6] for evolution of polygonal curves or polyhedral surfaces by crystalline curvature flow.
When we consider evolution of polygonal spirals with (3) with U = 0, then we are faced to the following characteristic problem: • Does the center of the spiral move or not?
• Generation of new facets from the center of the spiral. As the simple example, we now consider the situation such that a straight line with infinite length evolves by (3) with U > 0. If the center moves associating with the facet, then the "spiral" does not evolve since the line just move to the normal direction. Then, for the "evolution of spiral", one can find that the center should stay around the initial location with generating new facets so that the polygonal curve forms a spiral (see Figure 1 for the illustration of the generation of new facets). Imai, Ishimura and Ushijima [9] proposed a formulation for evolving polygonal spirals with the idea by [22] without generation of new facets. However, they investigate the evolution of curves only with U = 0, then they are concerned on the extinction of facets. On the other hand, the first author [13] proposed the evolution of spirals with generation of new facets. In this theory the center of spiral is moved on a pre-determined polygonal trajectory of the center corresponding with W γ , and then a new facet is generated when the center go through the corner on the trajectory.
In this paper, we propose a new formulation of evolving polygonal spiral by (3) with pinned center at the origin. In our algorithm a new facet should be generated when the facet associated with the center has enough long length for the facet of W γ which has the same direction as the evolving facets. Then, as mathematical analysis of our scheme, we consider not only the existence and uniqueness of solutions to the system of facet length, but also the existence of an infinite time sequence {T n } at when a new facet generates. Moreover, we show an important property for evolution of spiral: lim n→∞ T n = ∞ under a suitable setting of admissible initial curve. We also prove that the evolving curve has no self-intersections. As results in this paper, a polygonal spiral generates new facets repeatedly and then spiral steadily grows.
If the evolution is isotropic or anisotropic with smooth and convex density, then we have several PDE approach for evolution of spirals. It is well-known that phase-field method or level set method is powerful option to describe the motion of interfacial curve by (3). Then, there are several developments to apply their method to the evolution of spirals. Karma-Plapp [15], Kobayashi [16], and Miura-Kobayashi [17] proposed a formulation of evolving spirals by phase-field method with multiple-well potential and a pre-determined function (called sheet structure function) reflecting the sheet structure of the crystal lattice. On the other hand, Smereka [20] proposed a level set method with two auxiliary functions. The second author also proposed a level set formulation with a single auxiliary function and the sheet structure function due to [16] for single or several evolving spirals in [18] or [19]. However, the formulations in [15,16,18,19,17] requires to remove an open neighborhood of each center from the domain, since their equation have strong singularity at each center. In other words, their methods regard the centers of spirals as the open neighborhoods. Forcadel, Imbert and Monneau [3] give a formulation for a single evolving spiral with a pinned center as the origin. This paper is organized as follows. We first prepare some notations and definitions in §2. As the new feature, we divide the definition of admissibility as in [13] into semiadmissibility (only the continuity of the direction of facets), and admissibility (semiadmissibility with intersection free). We also define the generation time and the rule of the generation of new facet. Then, we introduce a scheme of the evolution of spirals, and define its semi-solution (solution of the length system with semi-admissibility) and solution (the solution with admissibility).
According to the definition defined in §2, we prove the existence and uniqueness of solutions of our scheme in §3. We divide the proof into two parts. The first part is the existence and uniqueness of solution to the length system, which is presented in §3.1. In the proof of existence, the continuity of the direction of facets are derived automatically. The second part is the intersection free property. However, the rigorous proof of the intersection-free property is established with just a precise calculation of inner products of some vector investigating the direction of facets, or detection of interior or exterior of a domain whose boundary is hyperplane. Thus, we just present an idea of the proof in §3.2. The rigorous proofs are presented in Appendix ( §5.2).
We also present some numerical results in §4. In §5, we mention on some remarks of our scheme. In §5.1 we mention on how to determine the direction of new facets. In §5.2, we give some rigorous proof of some properties on intersection.
2.1. Preliminaries. We prepare some notations and assumptions. We now recall a Wulff shape of an anisotropic surface energy. Let W γ be a set defined by (4), which is called Wulff shape of the surface energy defined by (2) with a continuous density function γ : S 1 → (0, ∞).
We here impose that W γ is a N γ sided convex polygon. The j-th facet of W γ has an outer unit normal vector N j with angle ϕ j for j ∈ Z/(N γ Z), and set the unit tangential vector T j of the j-th facet as well as the definition of the Frenet frame, i.e., N j = (cos ϕ j , sin ϕ j ), T j = (sin ϕ j , − cos ϕ j ).
We here consider a generalized number of facets j ∈ Z/(N γ Z) i.e., we regard j + nN γ as j for every n ∈ Z. From the convexity of W γ we assume that (W1) 0 = ϕ 0 < ϕ 1 < ϕ 2 < · · · < ϕ Nγ −1 < 2π, (W2) (ϕ j <)ϕ j+1 < ϕ j + π for every j ∈ Z/(N γ Z). We denote the length of the j-th facet of W γ by ℓ j > 0. We next prepare some notations for an evolving spiral-shaped polygonal curve by a crystalline curvature flow with respect to W γ . Let Γ(t) be a piecewise linear curve which has k + 1 facets denoted by L j (t) for j = 0, 1, 2, . . . , k; set Γ(t) = k j=0 L j (t). The j-th facet L j (t) is given as with the center of Γ(t) denoted by y k (t), vertices denoted by y j (t) for j = 0, 1, . . . , k−1 of Γ(t), and a given unit tangential vector τ 0 ∈ S 1 of L 0 (t). Let be the unit tangential vector of L j (t) j = 1, 2, . . . , k. We now impose that, for j = 0, 1, 2, . . . , k, there exists ν(j) ∈ Z/(N γ Z) such that We call ν(j) on the above a corresponding number of j-th facet to W γ . Thus, we now remark that L j (t) has other description as We denote the direction of the evolution of Γ(t) by the unit normal vector n j ∈ S 1 of L j (t) for j = 0, 1, 2, . . . , k. For each curve Γ(t) we set an orientation coefficient α ∈ {±1} such that (8) n j = αN ν(j) for j = 0, 1, 2, . . . , k provided that (6) holds. From the above context we have for s j (t) = y j (t) · n j , whereṡ j denotes the time derivative of s j (t). The function s j is called support function of L j (t). Note that L j (t) ⊂ {x ∈ R 2 ; x · n j = s j (t)}. Evolving spiral has a rotational orientation with respect to the center provided that its normal velocity is positive. See [19] for the definition of rotational orientations on the evolving smooth curve. We now introduce them to the evolving polygonal curve. We also introduce some geometric properties of Γ(t). See also [13]. Definition 1.
be an admissible spiral with respect to W γ , and oriented with the coefficient α ∈ {±1}, and assume that n j = αN ν(j) with ν(j) ∈ Z/(N γ Z). We define the crystalline curvature H j of an intermediate facet L j (t) of Γ j (t) with respect to W γ as where d j = |y j − y j−1 | is the length of L j , and ℓ ν(j) is the length of the ν(j)-th facet of W γ for j ∈ Z/(N γ Z).
2.2. Evolution system and scheme. Let Γ(t) = k j=0 L j (t) be an admissible spiral evolve with the normal velocity V j for j = 0, 1, 2, . . . , k. We now introduce a scheme of the evolution of Γ(t) by (3) with a crystalline surface energy and the pinned center at the origin, and a rule of the generation of a new facet at the center of Γ(t). Note that each L j (t) is given as (5), and then the center of Γ(t) is y k (t), which is fixed at the origin. Then, we impose that This condition should be taken over to a new facet L k+1 when it is generated. We now consider only the evolution of a positive spiral to avoid complication of formulation, then we set α = 1. We also may assume that U > 0. (The case when α = −1 or U < 0 will be mentioned later.) We shall use the same hypothesis on §2.1, then we observe that n j = N ν(j) , τ j = T ν(j) for j = 0, 1, 2, . . . , k with the corresponding facet number ν(j) defined in (6) for j = 0, 1, 2, . . . , k.
We first derive the evolution system of Γ(t) by (3) with a crystalline surface energy. According to [8], the evolution of Γ(t) with the normal velocity V j of L j (t) is described byḋ where d j (t) = |y j (t) − y j−1 (t)| (See also the Figure 4). If Γ(t) evolves by (3) with the crystalline energy, then V j is given as where β j = β(n j ) and H j is the crystalline curvature with respect to W γ defined in Definition 2. Note that L 0 (t) has infinite length as in (5), and then we regard H 0 = 0. By combining above and (9) we now obtain the system of the ordinary differential equations on the length d j (t) of L j (t); where b j , c ± j ∈ R are constants given by for j = 0, 1, 2, . . . , k. We solve the system (11) at least where d j > 0 for j = 1, 2, . . . , k − 1, and describe Γ(t) with setting to obtain the time-local evolution of Γ(t). Note that first formula in (11) is formally out of the system. However, d k has a role deciding a time of the generation of a new facet explaned below. For the evolution of a "spiral" generation of a new facet at the center of Γ(t) is necessary. In this paper we give a rule of the generation as resultant of the evolution of the present facet associated with the center. Let T k ∈ R be a time when L k (t) is generated, i.e., d k (T k ) = 0 and τ k = T ν(k) is given. We define a generation time of L k+1 (t) as We summarize the generation rule of a new facet as follows.

Summary of the scheme(SP)
Step 1. Give an suitable initial curve Γ( and obtain the solution (5) with y k (t) = O and y j (t) given by (12) for j = k, k − 1, . . . , 0.
Step 4. If T k+1 < ∞, where T k+1 is defined as (13), then generate L k+1 with the rule (G). and y k+1 = O at t = T k and return to Step 2 with updating the number of facets from k to k + 1, and initial time T k+1 .
We define the some classes of solution Γ(t) evolving by (3) with a crystalline surface energy.
is the life span of the system (11). 3. Global existence and admissibility. In this section we prove the existence and uniqueness of a global solution Γ(t) evolving by (3) with a pinned center. The main result of this section is as the following. Throughout this section we assume that U > 0 and consider an evolving positive spiral.
3.1. Existence and uniqueness. To prove Theorem 5 we divide the proof into two steps; the existence of semi-solution and intersection-free result. In this subsection we verify the following existence result on semi-solution as the first step. We first consider the local semi-solution to the system (11) for t ≥ T k with fixed k ≥ k 0 and a positive convex semi-admissible Γ(T k ). Note that Γ(T k ) is positive, then we may assume that (14) ν(j) = j, i.e., n j = N j without loss of generality. Moreover, by the convexity of Γ(T k ), the system (11) is simplified aṡ with constants b j , c ± j ∈ R at least in a short time for t ≥ T k . Note that c ± j > 0 by (14). We now demonstrate that the solution (d 1 , . . . , d k−1 , d k ) to (15)-(16) exists uniquely and globally-in-time provided that d 1 ( where κ j = ℓ j /δ j provided that j = 1, 2, . . . , k − 1, and κ k = U , κ 0 = 0. Moreover, we also demonstrate that the facet L k+1 generates at a finite time T k+1 < ∞, and d j (T k+1 ) orḋ j (T k+1 ) for j = 1, 2, . . . , k keep (I2i) and (I2ii). The result on the above should be summarized as the following.
Proof. We first note that the k = 1 is clear since this case is jusṫ For the case k ≥ 2 we divide the proof into the three steps.
Step 1. We first demonstrate that there exists a solution (d 1 , . . . , d k−1 ) to (16) (17) and (18), which implies (i) and (ii). For this purpose we represent (16) aṡ with d = (d 1 , . . . , d k−1 ) and F : Q := k−1 j=1 [ℓ j /(2U ), ∞) → R k−1 denoting the righthand side formula of (16). Then, since F is bounded and Lipschitz continuous on (20) for a constantρ > 0 by the usual iteration in the theory of ordinary differential equations. Moreover, the constantρ > 0 is independent of δ ∈ Q 0 . Then, we now demonstrate that for j = 1, 2, . . . , k − 1 to derive the existence of the global solution. In fact, we can extend d to the solution on [T k , T k + 3/2ρ] by solving (19) with an initial time t = T k +ρ/2 and an initial data δ = d(T k +ρ/2). The above implies that we can extend the solution to (19) To prove (21) set Note thatT is well-defined by (18) for every j = 1, 2, . . . , k − 1. The above properties implyḋ j0 (T ) ≤ 0, and thus j 0 = 1. In fact, if j 0 = 1 then we have by the third formula of (16), which is the contradiction. Thus we obtain d 1 (T ) > ℓ 1 /U , which yields j 0 = 2. In fact, if j 0 = 2, then by the second formula of (16), which is the contradiction. Then, we obtain j 0 / ∈ {1, 2, . . . , k − 1} by the inductive argument of the above, which contradicts to the definition ofT . Hence, we obtain (21), which implies the existence of the global solution to (19)- (20). Moreover, we obtain which are (i) and (ii) by the same argument of the above (for (i)) and the straight forward calculation (for (ii)).
By the result of Lemma 7 we obtain the following monotonicity principle.
Corollary 8. Let Γ(T k0 ) be a positive spiral curve satisfying (I1) or (I2). Let s j (t) = y j (t) · N j be a support function of L j (t) for t ≥ T j , where T j is the generation time of L j (t) and accordingly T j = T k0 if j ≤ k 0 . Then, the followings hold.
(ii) s i (t) = s j (t) = 0 provided i < j if and only if i + 1 = j ≥ k 0 and t = T j .
We omit the proof of Corollary 8 since it is obtained from the straightforward calculation ofṡ j (t) =ẏ j (t) · N j . We are in the position to prove Theorem 6.
Proof of Theorem 6. As we already mentioned in this subsection, we may assume that ν(j) = j. We divide the proof into 2 steps. We shall demonstrate that there exist the generation time T k and d k = (d k,1 , . . . , d k, (17)- (18) with δ j = δ k,j := d k,j (T k+1 ) and κ j = κ k,j := ℓ j /δ k,j for j = 1, 2, . . . , k − 1 in Step 1 by inductive argument. Then, we obtain the result of Theorem 6 in Step 2 by combining d k .
Step 2. We now construct a semi-solution Γ(t) for t ≥ T k0 . Set for k ≥ k 0 and j ∈ N where T j is what we obtained in Step 1 for j > k 0 , and accordingly T j = T k0 if j ≤ k 0 . Note that d j ∈ C 1 [T j , ∞). In fact, d j ∈ C[T j , ∞) and d j ∈ C 1 (T k , T k+1 ) for k ≥ k 0 by its definition and the property (ii) of d k . Then, it suffices to prove in particular for k = j + 1, j + 2 to prove d ∈ C 1 [T j , ∞), since the equation of d j for j ≥ k 0 − 1 is exactly changed at t = T j+1 and t = T j+2 ; see (15) and (16). We now verify the above for T j+1 . On one hand, by (15). On the other hand, by the first equation of (16) and the property (ii) and (iii) of d k , i.e., d j+1,j (T j+1 ) = ℓ j /U and d j+1,j−1 (T j+1 ) = d j,j−1 (T j+1 ). Hence, we obtain (22) at t = T j+1 , which implies the differentiability and continuity ofḋ j at t = T j+1 . We also obtain (22) at t = T j+2 by the similar argument with d j+2,j+1 (T j+2 ) = ℓ j+1 /U . We now construct Γ(t) for t ≥ T k0 as a family of polygonal curves in each interval I k ; set Γ(t) := k j=0 L j (t) if t ∈ I k for each interval I k for k ≥ k 0 with L j (t) given by (5) and y k (t) := O, y j (t) := y j+1 + d j (t)T j for j = k − 1, k − 2, . . . , 0.

Self-intersection free.
In the previous section we prove the existence of a semi-solution Γ(t) provided that initial curve Γ(T k0 ) satisfies (I1) or (I2i)-(I2ii). We shall prove the admissibility of Γ(t) to obtain the solution to (3) with scheme (SP). Note that the properties (A1) and (A2) of the admissibility are guaranteed by Lemma 7 and the generation rule (G). Then, we now prove that Γ(t) is self-intersection free for t ≥ T k0 .
To verify the self-intersection free result, it is convenient to introduce semi-open and open line segment description of L j (t) as (5) to Γ(t) = k j=0 L j (t). Define Then, we observe that k j=0 L j (t) = k j=0 Λ j (t). We now classify the kinds of selfintersection of a polygonal curve from the usual definition of self-intersection.
Definition 9. We say Γ(t) = k j=0 L j (t) = k j=0 Λ j (t) has a self-intersection if there exist i, j ∈ {0, 1, . . . , k} such that i = j and Λ i (t) ∩ Λ j (t) = ∅. Moreover, we classify the kinds of self-intersections between Λ i (t) and Λ j (t) as the following; (  5. Examples of self-intersection between Λ i (t) (solid line) and Λ j (t) (dashed line). Note that dots means y i (t) or y j (t), which is a vertex belongs to Λ i (t) or Λ j (t), respectively.
Then, we obtain the following self-intersection free result.
Lemma 11. Assume thatt < ∞ and a self-intersection appears between Λ i (t) and Λ j (t) with i < j. Then the following properties hold.
(i) The intersection between Λ i (t) and Λ j (t) is not the cross intersection.
(ii) If the intersection between Λ i (t) and Λ j (t) is the facet-facet intersection, then and Λ j+1 (t) are not empty in a neighborhood oft.
Proof. We first demonstrate (i). Let us choose We fix the small constantμ > 0 and the open neighborhood U of y satisfying We may assume that z j (t, r − ) · N i < s i (t) and z j (t, r + ) · N i > s i (t) without loss of generality. Then, there exists µ ∈ (0,μ) such that for some r 0 ∈ (r − , r + ) by (24), which contradicts to the definition oft. Hence, we obtain the conclusion (i).
We next verify (ii). Assume that the facet-facet intersection appears between Λ i (t) and Λ j (t) with N i = −N j . Then, we observe that s i (t) = −s j (t) = y · N i with some y ∈ Λ i (t) ∩ Λ j (t). Thus, if s i (t) = 0, then s i (t)s j (t) < 0 which contradicts to s i ≥ 0 and s j ≥ 0 on [T k0 , ∞) by Corollary 8. This yields that s j (t) = s j (t) = 0. However, this situation appears only whent = T j by (ii) in Corollary 8. This implies Λ j (t) = ∅ which contradicts to the definitions of the self-intersections. Hence, we obtain (ii).
We demonstrate (iii). Note that j ≥ i + 1 > i and thus each Λ j (t), Λ i+1 (t) or Λ i (t) is not empty. Then, we now lead a contradiction with assuming Λ j+1 (t) = ∅. In fact, this implies thatt ∈ (T j , T j+1 ] by (i) in Corollary 8, and then y j (t) = O. We now remark that the situation of the intersection between Λ i (t) and Λ j (t) should be divided into the following three cases; • facet-facet intersection with N i = N j (see (ii)), However, we observe that s i (t) = s j (t) = 0 for each above case. In fact, for the first case of the above we find s j (t) = y j · N j = y · N j = 0 for some y ∈ Λ i (t) ∩ Λ j (t), and then s i (t) = y · N i = y · N j = 0. For the second and third cases we observe that O ∈ Λ i (t) ∩ Λ j (t), which implies that s i (t) = s j (t) = 0. Thus, we observe that j = i + 1 witht = T i+1 by (ii) in Corollary 8, which implies that Λ j (t) = ∅. This is the contradiction. Hence, we obtain (iii).
Proof of Theorem 10. We here mention only the sketch of the proof since the all observation are derived from elemental calculations of inner product and continuity of y j (t). (See §5.2 for the details.) Lett ∈ [T k , T k+1 ) be a first touch time for Γ(t) = k j=0 Λ j (t), and let Λ i (t) ∩ Λ j (t) = ∅. We may assume that without loss of generality by choosing suitable n ∈ Z and ϕ j + 2πn instead of ϕ j if necessary. We shall prove the followings. (I) If Λ i (t) and Λ j (t) has facet-vertex type intersection with Λ • i (t) ∩ Λ j (t) = {y j (t)}, then Λ i (t) and Λ j+1 (t) has facet-facet type intersection. (II) If Λ i (t) and Λ j (t) has vertex-vertex type intersection, then the both pair (Λ i (t), Λ j (t)) and (Λ i+1 (t), Λ j+1 (t)) have facet-facet type intersection. (III) If Λ i (t) and Λ j (t) has facet-facet type intersection, then Λ i (t) = Λ j (t) with N i = N j . Then, if (I), (II) and (III) are valid, then there exists a pair of facets having facet-facet type intersection, which is still denoted by Λ i (t) and Λ j (t) with i < j for the simplicity. Then, we obtain Λ i (t) = Λ j (t) by (III), which implies that in particular Λ i−1 (t) and Λ j−1 (t) has facet-facet type intersection. This also implies that Λ i−1 (t) = Λ j−1 (t). By iterating the above argument in finite times we obtain Λ 0 (t) = Λ j−i (t). This contradicts to the fact that Λ 0 (t) is unbounded for t ≥ 0. According to the above argument, it suffices to verify (I), (II) and (III).
(I) See the proof of Proposition 12 in §5.2 for details.
Note that Λ j+1 (t) = ∅ in this case. In fact, it follows from Lemma 11(iii) if i < j. On the other hand, let i > j, which implies i ≥ j + 1, and assume that Λ j+1 (t) = ∅. Then, we find i = j + 1 and thent = T i by (ii) in Corollary 8. This contradicts to Λ i (t) = ∅.
We divide this case into two cases; Figure 6), then we observe that Λ j+1 (t) ⊂ {x ∈ R 2 ; x · N j < s j (t), and x · N i > s i (t)} (the gray region of the figure (a)). However, the above implies and then y j (t) = O by (i) in Corollary 8. This implies s i (t) = s j (t) = s j+1 (t) = 0, which contradicts to (ii) in Corollary 8 and (iii) in Lemma 11. If Λ • j (t) ⊂ I i (t) (see Figure 7), then we observe that (the gray region of the figure (a)). In this situation Λ j+1 (t) only can be located on L i (t) := {x ∈ R 2 ; x · N i = s i (t)} by the property (A2) in Definition 1 of Γ(t), which is the conclusion of (I). In fact, if Λ j+1 (t) ⊂ I j (t), then we observe that  for i, j, j + 1 ∈ Z/(N γ Z). This contradicts to the assumption (W1).
(III) Assume that Λ i (t) = Λ j (t). When one is included by the other, we may assume Λ i (t) ⊂ Λ j (t) without loss of generality. In this case we find i = 0. In fact, if i = 0, then Λ j (t) is also unbounded, which contradicts to d j (t) < ∞ for j ≥ 1. Moreover, This contradicts to the definition oft.
It remains the case when Λ i (t) ⊂ Λ j (t) and Λ i (t) ⊃ Λ j (t). If i ≥ 1 and j ≥ 1, then we may assume that there exist λ 3 > λ 2 > λ 1 > 0 such that i.e., y j (t) ∈ Λ • i (t) and y i−1 (t) ∈ Λ • j (t). In this case there exists µ > 0 such that either the following (a) or (b) holds; (a) s i < s j on [t − µ,t), Figure 9.) with constants λ 2 > λ 1 > 0. Moreover, we observe thatṡ j (t) = V 0 >ṡ i (t). Thus, we lead a contradiction with the same argument of the case (b) on the above. We thus obtain the conclusion of (III), and accordingly that of Theorem 10.

Numerical simulations.
In this section we consider the evolution equation instead of (3) for the consistency with [1]; v ∞ > 0 and ρ c > 0 are constants denoting the mobility of the evolution and the critical radius of the equilibrium form, respectively. One can easily obtain a solution to (3) (whose vertices are denoted by y j (t)) from that to (29) (whose vertices are denoted by η j (τ )) by the scheme (SP) by rescaling the time and spatial paramters; set In this section we only consider the initial curve of spiral as that satisfying (I1), and then the solution should be a positive convex spiral by Theorem 5. Consequently, we calculate the system (16) adapting the coefficients to (29). For example, the third equation of (16), which is typical one, should be revised aṡ and the other equations should be revised as the same manner. Note that ρ c W γ = {ρ c x ∈ R 2 ; x ∈ W γ } is a stationary solution of closed curve to (29), and then the critical length of j-th facet evolving by (29) is ρ c ℓ j . One can easily calculates the system (16) for (29) with an usual explicit finite difference scheme. The figure 10 presents a numerical result of the evolution of a polygonal spiral at time t = 0, 0.1, 0.5 and 1.0 with triangle W γ and isotropic β. We set the parameters of W γ as The parameters of (29) are β ≡ 1, v ∞ = 1, ρ c = 0.02. For evolution of closed curve, Yazaki [23] shows the asymptotic behavior of solution to the Wulff shape for the anisotropy of βV = 1. In the context of this paper one was afraid that the facets surrounded by other facets having relatively small mobility (and then large velocity) might be vanished. However, a priori estimates in Lemma 7 or Corollary 8 guarantee that the all facets never be vanished and evolve with positive normal velocities. On the other hand, in particular the estimate (iii) in Lemma 7 does not guarantee that lim t→∞ d j (t) = ∞; see case (d) in Figure 11.

Appendix.
5.1. Remark on the orientation of a new facet. We now remark on the rule of the orientation of generated facet in the generation rule (G) in §2.2. Formally, there exists a possibility setting ν(k + 1) = ν(k) − 1 when L k+1 (t) is generated at t = T k+1 provided that either Γ(t) is positive and U > 0 or Γ(t) is negative and U < 0. We here mention why we can avoid the above situation. Note that the case for negative Γ(t) with U > 0 (or positive Γ(t) with U < 0) is considered in parallel.
If we can avoid the assumptions of Γ(t) in (I1) or (I2i)-(I2ii), one attempt to consider the evolution of positive and convex Γ(t) = k+1 j=0 L j (t) with an initial data Γ(T k+1 ) satisfying If L k+1 (t) is oriented as ν(k+1) = ν(k)+1, then one can find d k+1 < 0 in (T k+1 , T k+1 + µ) with a small µ > 0. Thus, if we keep setting y k (t) = y k+1 (t) + d k+1 (t)τ k+1 in spite of the fact that d k+1 (t) < 0, the admissibility of Γ(t) is broken (see Figure 12(a)). It is natural for the above case that one attempt to set ν(k + 1) = ν(k) − 1 to keep the admissibility of Γ(t). (See figure 12(b).) However, if we set ν(k + 1) = ν(k) − 1, then we have σ k = 0, and then the evolution equation for L k (t) should be β k V k = U , which implies s k (t) > 0 in a very short time from T k+1 . Moreover, we observe thaṫ Consequently, there is no way to set ν(k + 1) without breaking the admissibility of Γ(t) for the above toy case. It is very important to guarantee d k ≥ ℓ ν(k) /U for t > T k+1 provided that Γ(t) = k+1 j=0 L j (t) is convex and U > 0, and then we can avoid the case ν(k + 1) = ν(k) − 1 under the above guaranty.

Proof of Theorem 10.
In this section we give a proof on the observations (I), (II) and (III) in the proof of Theorem 10. We assume that Γ(t) is positive convex, and then (14) holds. Throughout this section we consider Γ(t) = k j=0 Λ j (t) has a first touch timet < ∞ which is defined as (23), and Λ i (t)∩Λ j (t) = ∅. We may assume that (25), i.e., ϕ i ≤ ϕ j < ϕ i + 2π without loss of generality. Moreover, we also note that Λ i+1 (t) and Λ j+1 (t) exist in a neighborhood oft by Lemma 11 (iii). We first demonstrate the observation (I) on the facet-vertex type intersection; when the facet-vertex type intersection appears, then the adjacent facets have the facet-facet type intersection.
We next deduce a contradiction from (31). By straightforward calculation we observe that This implies that x · N i ≥ s i (t) for x ∈ I j (t) ∩ I j+1 (t), and Hence, we obtain I i (t)∩I j (t)∩I j+1 (t) = {y j (t)}. Note that O ∈ I i (t)∩I j (t)∩I j+1 (t) by Corollary 8, which implies s i (t) = s j (t) = s j+1 (t) = 0 and contradicts to Corollary 8 (ii).
We next demonstrate the observation (II); if the vertex-vertex type intersection appears, then the two pair of facets associated with the touched pair of vertices have facet-facet type intersections.
Proof. It suffices to prove N i = N j , and thus ϕ i = ϕ j provided that (25) holds. We now assume that ϕ i = ϕ j and derive a contradiction.
Finally, we demonstrate the observation (III); if the pair of facets has facet-facet type intersection, then the facets are agree with each other.
Proposition 14. If Γ(t) has a facet-facet type intersection between Λ i (t) and Λ j (t), then Λ i (t) = Λ j (t) with N i = N j .
Proof. Note that N i = N j follows from Lemma 11(ii). Then, we now lead a contradiction with assuming Λ i (t) = Λ j (t). By definition of s j (t) we have y j · N j = y j−1 · N j = s j , and we also have y j · N i = y j−1 · N i = s j since N i = N j . Moreover, by assumption we now have y j (t) · N i = y j−1 (t) · N i = s i (t) = s j (t).
We also note that each Λ i+1 (t) or Λ j+1 (t) is not empty in a neighborhood of t =t by Lemma 11(iii).
We first note that I i+1 (t) ∩ I i−1 (t) ∩ L i (t) = Λ • i (t) for t ≥ T i+1 provided that the set on the left hand is not empty, where L i (t) = {x ∈ R 2 ; x · N i = s i (t)}.
In fact, we have (43) with r ≤ 0 or r ≥ d i (t) by the assumption. If r ≤ 0, then we obtain x · N i+1 ≤ s i+1 (t) by the first equality of (44), which implies x / ∈ I i+1 (t). On the other hand, if r ≥ d i (t), then we obtain x · N i−1 ≤ s i−1 (t) by the first equality of (45), which implies x / ∈ I i−1 (t). Hence, we obtain Λ • i (t) = I i+1 (t) ∩ I i−1 (t) ∩ L i (t). We now divide the proof of Proposition 14 into two cases.
Step 1. Consider the case Λ j (t) ⊂ Λ i (t) or Λ j (t) ⊃ Λ i (t). We may assume that Λ j (t) ⊂ Λ i (t) by switching the number i and j if necessary. Note that j = 0. In fact, if j = 0 then Λ i (t) and Λ j (t) are unbounded, which contradicts to d i < ∞ for i ≥ 1. Since Λ i (t) = Λ j (t), we observe that y j ∈ Λ • i (t) or y j−1 ∈ Λ • i (t). Since the proofs are parallel, we now consider only the case when y j ∈ Λ • i (t). The above assumptions imply d j (t) < d i (t). Then, we haveṡ j (t) <ṡ i (t) sincė s m = V m = β −1 m (U − ℓ m /d m ) for m ≥ 1 and N i = N j implies j = i + nN γ with n ∈ Z. Then, there exists µ 0 > 0 such that (46) s j (t) < s i (t) for t ∈ [t − µ 0 ,t).
Step 2. We next consider the case Λ i (t) ⊂ Λ j (t) and Λ i (t) ⊃ Λ j (t). We first consider the case when i ≥ 1 and j ≥ 1. Then, either the following (A) or (B) holds.