Ishiwata , Tetsuya

EPrint Series of Department of Mathematics, Hokkaido University

Behavior of solutions to an area-preserving crystalline motion
Tetsuya ISHIWATA ¤
joint work with Shigetoshi YAZAKI(University of Miyazaki)
1 Introduction
When a block of ice crystal is illuminated by strong beams, the ice crystal starts to melt
inside of the crystal as well as the surface and each water region forms a snow°ake-like-
pattern which has six petals, called \Tyndall ¯gure" (see Figure 1 (a)). This ¯gure has a
vapor bubble in water region and when this ¯gure is refrozen, the vapor bubble remains in
the ice as a hexagonal disk (see Figure 1 (b)). This hexagonal disk is a kind of negative
crystals and the interior region is ¯lled with water vapor saturated at that temperature.
McConnel([6]) found these disks in the ice of Davos lake. Nakaya called this hexagonal disk
\Kuuzou(空像)" in Japanese and investigated its properties [7].
YX
KEG
YCVGTXCRQTYX
(a) (b)
Figure 1: (a) Tyndall ¯gures (seen from 45± to the c-axis) and (b) a negative crystal (by U.
Nakaya).
In [5], we proposed a motion equation for a polygonal curve in the plane as a simple
model of the formation process of negative crystals after the water region in a Tyndall ¯gure
is completely refrozen. This model equation is obtained by a gradient °ow of total surface
energy under an area-preserving constraint:
Vi = H ¡Hi:
Here Vi is the outward normal velocity on the i-th facet Fi of vapor region ­(t) (enclosed
region by a polygon), Hi is the crystalline curvature of Fi and H is the average of all
¤Department of Mathematical Sciences, College of Systems Engineering and Science, Shibaura Institute of
Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 337-8570, JAPAN. E-mail: tisiwata@shibaura-
it.ac.jp
1
crystalline curvatures. This equation is called area-preserving crystalline motion or area-
preserving crystalline curvature °ow. Crystalline motion is a singular weighted curvature °ow
with non-smooth surface energy ° and J. Taylor[8] and Angenent and Gurtin [1] proposed
the framework of crystalline motions. In this framework, the interfaces are restricted in
the class of polygonal curves (two dimensional case) which satisfy an admissibility condition
based on the equilibrium shape of the crystal. This equilibrium shape is called the Wul®
shape and plays important roles for not only the de¯nition of the crystalline curvature and
admissibility condition, but also the asymptotic behavior of the solution polygons. The
detailed formulations will be mentioned in next section.
In the case that an initial shape ­0 is convex, the solution polygon ­(t) keeps its con-
vexity. S. Yazaki [9, Part I] show that no facets disappear globally in time and the solution
polygon converges to the rescaled Wul® shape whose area is equal to that of ­0 in the Haus-
dor® metric. However, when the vapor region is surrounded by the ice region in refreezing
process, many ¯ne facets appear on the interface and the shape of the vapor region is not
convex in general. Thus, in this talk, we consider the case that ­0 is not convex. In this
case, there is a possibility that the solution has some singularities in ¯nite time, for exam-
ple, facet-extinction and self-intersection of the interface. We show the su±cient conditions
on the Wul® shape and an initial polygon to keep admissibility of the solution polygons.
Moreover, we also show that the solution polygon from non-convex initial polygon becomes
convex in ¯nite time.
2 Area-preserving crystalline motion
Crystalline energy and the Wul® shape. Let ° = °(n) be a positive continuous function
de¯ned on S1 and describe interfacial energy density for the direction n. In this note, we
consider the case where the Wul® shape of °, W° = fx 2 R2jx ¢ n · °(n) for all n 2 S1g,
is a convex polygon. Such ° is called crystalline energy. If W° is a J-sided convex polygon
(J ¸ 3), then W° is expressed as
W° =
J\
i=1
©
x 2 R2; x ¢ ºi · °(ºi)
ª
;
where ºi = n(Ái) and Ái is the exterior normal angle of the i-th facet with Ái 2 (Ái¡1; Ái¡1+
¼) for all i (Á0 = ÁJ , ÁJ+1 = Á1). We de¯ne a set of normal vectors of W° by N° =
fº1;º2; : : : ;ºJg.
Polygons and polygonal curves. Let ­ be N -sided polygon in the plane R2, P its bound-
ary, that is, P = @­ and label the position vector of vertices pi (i = 1; 2; : : : ; N) in an
anticlockwise order: P = SNi=1Fi, where Fi = f(1¡ t)pi + tpi+1; t 2 [0; 1]g is the i-th facet
(p0 = pN , pN+1 = p1). The length of Fi is di = jpi+1 ¡ pij, and then the i-th unit tangent
vector is ti = (pi+1 ¡ pi)=di and the i-th unit outward normal vector is ni = ¡t?i , where
(a; b)? = (¡b; a). We de¯ne a set of normal vectors of P by N = fn1;n2; : : : ;nNg. Let µi be
the exterior normal angle of Fi. Then ni = n(µi) and ti = t(µi) hold (µ0 = µN , µN+1 = µ1),
where t(µ) = (¡ sin µ; cos µ).
We de¯ne the i-th hight function hi = pi ¢ ni = pi+1 ¢ ni (h0 = hN , hN+1 = h1). By
using fhi¡1; hi; hi+1g and fni¡1;ni;ni+1g, the length of i-th facet di is described as follows:
di =
Âi¡1;i(hi¡1 ¡ (ni¡1 ¢ ni)hi)p
1¡ (ni¡1 ¢ ni)2
+
Âi;i+1(hi+1 ¡ (ni ¢ ni+1)hi)p
1¡ (ni ¢ ni+1)2
; i = 1; 2; : : : ; N;
where Âi;j = sgn(ni ^ nj) and a1 ^ a2 = det(a1;a2) is the determinant of the 2£ 2 matrix
with column vectors a1, a2. Since ni ¢ nj = cos(µi ¡ µj), we have another expression:
di = ¡(cot#i + cot#i+1)hi + hi¡1 cosec#i + hi+1 cosec#i+1; i = 1; 2; : : : ; N; (1)
where #i = µi ¡ µi¡1. Note that 0 < j#ij < ¼ holds for all i. Furthermore, the i-th vertex pi
(i = 1; 2; : : : ; N) is described as follows:
pi = hini +
hi¡1 ¡ (ni¡1 ¢ ni)hi
ni¡1 ¢ ti ti: i = 1; 2; : : : ; N: (2)
Admissibility and crystalline curvature. We call ­ and P admissible (associated with
W°) if and only if N = N° holds and any adjacent two normal vectors in the set N are also
adjacent in the set N° , i.e., for any i, there exists j such that fºj ;ºj+1g = fni;ni+1g holds.
Let P be an admissible polygonal curve. For each facet Fi a crystalline curvature is
de¯ned by
H(Fi) = Âi l°(ni)
di
; i = 1; 2; : : : ; N;
where Âi = (Âi¡1;i + Âi;i+1)=2 is the transition number and it takes +1 (resp. ¡1) if P is
convex (resp. concave) around Fi in the direction of ¡ni, otherwise Âi = 0; and l°(ni) is
the length of the j-th facet of W° if ni = ºj . If ­ is an admissible convex polygon, then
ni = ºi and Âi = 1 for all i = 1; 2; : : : ; N = J ; and moreover, if ­ =W° , then the crystalline
curvature is 1. In this note, we call a facet which zero transition number \in°ection facet."
We note that the total interfacial crystalline energy on P is
E° =
NX
i=1
°(ni)di; (3)
and the crystalline curvature H(Fi) is characterized as the ¯rst variation of E° on P at Fi
with a suitable norm. Here and hereafter, we denote H(Fi) by Hi for short.
Area-preserving crystalline motion. The normal velocity on Fi in the direction ni is
Vi = _hi. Here and hereafter, we denote that the derivative of a function u = u(t) with respect
to time t by _u. The area-preserving crystalline motion is the gradient °ow of E° along P
which encloses a ¯xed area, and it is described as follows:
Vi = H ¡Hi; i = 1; 2; : : : ; N; (4)
where
H =
PN
i=1Hidi
L
is the average of the crystalline curvature, and L =PNk=1 dk is the total length of the curve
P. From (1), we have
_di = ¡(cot#i + cot#i+1)Vi + Vi¡1 cosec#i + Vi+1 cosec#i+1; i = 1; 2; : : : ; N: (5)
Furthermore, by (2) we have
_pi = Vini +
Vi¡1 ¡ (ni¡1 ¢ ni)Vi
ni¡1 ¢ ti ti; i = 1; 2; : : : ; N: (6)
Note that (4), (5) and (6) are equivalent each other. It is easy to check that the enclosed
area A(t) =PNi=1 hidi=2 is preserving in time: _A(t) =PNi=1 Vidi = 0.
3 Results
For any given admissible initial polygon ­0, we have short time existence and uniqueness
result by the standard argument since (5) is the system of ordinary di®erential equations.
Known results for convex polygons. What might happen to ­(t) as t tends to the
maximal existence time T · 1? For this question, we have the following result.
Theorem 1 Let the crystalline energy be ° > 0. Assume the initial polygon ­0 is an N -
sided admissible convex polygon. Then the solution admissible polygon ­(t) exists globally
in time keeping the area enclosed by the polygon constant A, and ­(t) converges to the shape
of the boundary of the Wul® shape @W°¤ in the Hausdor® metric as t tends to in¯nity, where
°¤(ni) = °(ni)=W , W =
pjW° j=A for all i = 1; 2; : : : ; N and jW° j = PNk=1 °(nk)l°(nk)=2
is enclosed area of W° .
This theorem is proved in Yazaki [9, Part I] by using the anisoperimetric inequality or BrÄunn
and Minkowski's inequality and the theory of dynamical systems.
Our results for non-convex polygons.
In the previous case, the solution polygon keeps its convexity and admissibility, that is,
the length of each facet is positive globally in time and the self-intersection of P(t) never
occur. However, if ­0 is non-convex, the facet-extinction or the self-touching may occur
in ¯nite time. Indeed, we can easily construct the example of the self-intersection of P(t)
and ­(t) becomes non-admissible after the singularity. Thus, the admissibility of solution
polygons may break down in ¯nite time. To track the motion globally in time in the class
of admissible polygons, we prepare the following assumptions:
(A1) W° is symmetric with respect to the origin:
(A2) Transition numbers of ­0 are all nonnegative: Âi ¸ 0 for any i:
Theorem 2 Assume the assumptions (A1) and (A2). Let ­0 be an N -sided non-convex
admissible polygon. Then, there exists T1 > 0 such that the solution polygon is an N-sided
admissible polygon for 0 · t < T1 and there exists at least one in°ection facet whose length
tends to zero as t ! T1. Moreover, ­(t) converges to an admissible polygon ­¤ in the
Hausdro® topology as t! T1 and area of ­¤ is equal to area of ­0.
This theorem means that we can restart the motion with the initial polygon ­¤ and
obtain the solution in the class of admissible polygons beyond the singularity. If ­¤ is non-
convex, then we can apply Theorem 2 again and again. We ¯nally have a ¯nite sequence
of facet-extinction time : 0 < T1 < T2 < ¢ ¢ ¢ < Tm < +1. Then, we obtain the following
convexity result.
Theorem 3 Assume that the same assumption as in Theorem 2. Then, the solution polygon
becomes convex at t = Tm.
After the convexity phenomena occurs, we can apply Theorem 1. Therefore, the solution
polygon exists globally in time in the class of admissible polygons and the solution polygon
¯nally converges to the rescaled Wul® shape.
4 For negative crystals
For usual crystal case, enclosed region describes the crystal and then normal vector n is
direction from the crystal to its outside region. However, for negative crystal case, the
outside region describes the crystal. Thus, applying the area-preserving crystalline motion
to understand the motion of the boundary of negative crystals, we need to use °(¡n) as the
interfacial energy density. Therefore, we use the ¯gure:
J\
i=1
©
x 2 R2; x ¢ (¡ºi) · °(ºi)
ª
;
as the Wul® shape for negative crystal case.
References
[1] S. Angenent and M.E. Gurtin, Multiphase thermomechanics with interfacial structure,
2. Evolution of an isothermal interface, Arch. Rational Mech. Anal. 108 (1989) 323{391.
[2] Y. Furukawa and S. Kohata, Temperature dependence of the growth form of negative
crystal in an ice single crystal and evaporation kinetics for its surfaces, J. Crystal Growth
129 (1993) 571{581.
[3] M. Gage, On an area-preserving evolution equations for plane curves, Contemporary
Math. 51 (1986) 51{62.
[4] M. E. Gurtin, Thermomechanics of evolving phase boundaries in the plane, Oxford,
Clarendon Press (1993).
[5] T. Ishiwata and S. Yazaki, Towards modelling the formation of negative ice crystals
or vapor ¯gures produced by freezing of internal melt ¯gures, RIMS Ko^kyu^roku 1542
(2007) 1{11.
[6] James C. McConnel, The Crystallization of Lake Ice, Nature, 39 (1889), 367.
[7] U. Nakaya, Properties of single crystals of ice, revealed by internal melting,
PIPRE(Snow, Ice and Permafrost Research Establishment) Research Paper 13 (1956)
[8] J. E. Taylor, Constructions and conjectures in crystalline nondi®erential geometry, Pit-
man Monographs Surveys Pure Appl. Math. 52 (1991) 321{336, Pitman London.
[9] S. Yazaki, On an anisotropic area-preserving crystalline motion and motion of nonad-
missible polygons by crystalline curvature, RIMS Ko^kyu^roku 1356 (2004) 44{58.


Behavior of solutions to an area-preserving crystalline motion

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EPrint Series of Department of Mathematics, Hokkaido University