Uniqueness and existence of generalized motion for spiral crystal growth

Abstract

Abstract. The uniqueness and existence of generalized solutions of 'spiral curves' for the mean curvature flow with driving force is studied by an adapted level set formulation. It is shown that the curves which are given by the level set formulation are unique with respect to initial spiral curves. For given spiral curves the method of a construction of an initial datum of a level set equation is also obtained by constructing a branch of the arguments from the centers of spiral curves, which has discontinuity at the given spiral curves. IntroductionVarious motions of spiral curves are observed on the growing surface of a crystal. This phenomena is come from spiral growth that occurs with aid of a screw dislocation intersecting the surface. In 1949, F. C. Frank pointed out important role of a screw dislocation to growth of a crystal in [5]. The theory of a growth of a crystal with aid of a screw dislocation is proposed by [1]. Recently this is called spiral growth of a crystal. According to the theory in [1], the spiral curves are described by steps on the surface of a crystal, and they move by the evolution law of the formwhere V is the normal velocity, κ is the curvature of the curve, and C is a constant Figure 1. A crystal surface.which denotes the driving force. This is an interface model of spiral crystal growth. We have two types of mathematical models of this phenomena, phase-field models or interface models. A. Karma and M. Plapp ([12]) proposed a phase-field model 2000 Mathematics Subject Classification. 53C44, 35K65.

Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

“…For given initial data u 0 ∈ C(W ), there exists a unique global viscosity solution u ∈ C([0, ∞) × W ) to ( 7)-( 8) with u(0, •) = u 0 . Moreover, the uniqueness of evolution of Γ L (t) has been established in [12]; if there are continuous viscosity solutions u and v to ( 7)-( 8) satisfying Γ u L (0) = Γ v L (0) with the same orientations, then Γ u L (t) = Γ v L (t) for t > 0, where Γ u L (t) = {x ∈ W ; u(t, x) − θ(x) ≡ 0 mod 2πZ}. Hence, we may give an arbitrary u 0 ∈ C(W ) to obtain the motion of Γ L (t).…”

Growth Rate of Crystal Surfaces with Several Dislocation Centers

“…However, the uniqueness of Γ t for a given Γ 0 is established provided that u 0 is continuous and the orientation of Γ 0 given by u 0 is the same. 23 In order words, Γ t depends only on Γ 0 and its orientation and is independent of the choice of the functions that embed it. Initial data u 0 for the simulations in this paper will be chosen as a constant or constructed from a union of lines: see the previous paper 7 for details of the construction.…”

“…In fact, ∇θ is well-defined on W̅ as for x = ( x 1 , x 2 ) and a j = ( a j ,1 , a j ,2 ) by taking a branch of θ so that it is smooth around x . Second, notice that u 0 satisfying eq is not unique even if u 0 is considered in the space of continuous functions. However, the uniqueness of Γ t for a given Γ 0 is established provided that u 0 is continuous and the orientation of Γ 0 given by u 0 is the same . In order words, Γ t depends only on Γ 0 and its orientation and is independent of the choice of the functions that embed it.…”

Steady State and Long Time Convergence of Spirals Moving by Forced Mean Curvature Motion

“…We point out that in [12], the geometric law is anisotropic, and is thus more general than (1.1). In [35,31,32,15], the geometric flow is studied by using the level-set approach. As in [12], the authors of [31,32] consider spirals that typically move inside a (compact) annulus and reaches the boundary perpendicularly.…”