DCDS-S
On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect
Tetsuya Ishiwata
The behavior of polygonal curves with asymptotic lines to crystalline motion with the bulk effect is discussed. We show sufficient conditions for global existence of the solutions and characterize facet-extinction patterns. We also show the eventual monotonicity of shape of the solution curves, that is, the solutions become V-shaped in finite time.
keywords: Eventual monotonicity. Crystalline curvature Crystalline motion
DCDS
A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion
Tetsuya Ishiwata Shigetoshi Yazaki
The asymptotic behavior of solutions to an anisotropic crystalline motion is investigated. In this motion, a solution polygon changes the shape by a power of crystalline curvature in its normal direction and develops singularity in a finite time. At the final time, two types of singularity appear: one is a single point-extinction and the other is degenerate pinching. We will discuss the latter case of singularity and show the exact blow-up rate for a fast blow-up or a type II blow-up solution which arises in an equivalent blow-up problem.
keywords: blow-up rate degenerate pinching blow-up core. Crystalline motion
DCDS-S
Crystalline motion of spiral-shaped polygonal curves with a tip motion
Tetsuya Ishiwata
In this paper we propose a crystalline motion of spiral-shaped polygonal curves with a tip motion as a simple model of a step motion on a crystal surface under screw dislocation. We give a tip motion and discuss the behavior of the solution curves by crystalline curvature flow with a driving force. We show that the solution curve belongs to a suitable class of spiral-shaped curves and also show a time-global existence of the spiral-shaped solutions.
keywords: behavior of solutions global existence. Motion by crystalline curvature polygonal curves spirals
PROC
Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field
Tetsuya Ishiwata Kota Kumazaki
In this short paper we propose a finite difference scheme for the Landau-Lifshitz equation and an iteration procedure to solve the scheme. The key concept is ``structure-preserving''. We show that the proposed method inherits important mathematical structures from the original problem and also analysis the iteration.
keywords: energy structure. fi nite difference scheme the Landau-Lifshitz equation length-preserving Structure preserving
DCDS-S
Preface: Special Issue on recent topics in industrial and applied mathematics
Michal Beneš Tetsuya Ishiwata Takashi Sakamoto Shigetoshi Yazaki
This issue consists of 16 carefully refereed papers dealing with mathematical modeling, numerical analysis, numerical simulation and mathematical analysis for ordinary/partial differential equations in several fields: fluid dynamics, chemistry, crystal physics, material science, medical science and biology. The authors in the present issue are mainly from Czech Republic, Slovakia and Japan. Scientific and mathematical exchanges began in 2004 among the authors and their friends. Since then, fruitful relationships and effective exchanges have been promoted and developed. Professor Tatsuyuki Nakaki strongly supported these exchanges and established the Czech-Slovak-Japanese joint seminars on industrial and applied mathematics. Unfortunately, he passed away in 2011. In 2013 in Tokyo, we had an opportunity to have a meeting in honor of Professor Nakaki. Recent topics in industrial and applied mathematics were widely discussed and we could exchange new ideas and various kinds of new problems. The present issue includes a part of those results.
    Finally we would like to thank the referees for their valuable comments in reviewing the papers.
keywords:
PROC
Motion of polygonal curved fronts by crystalline motion: v-shaped solutions and eventual monotonicity
Tetsuya Ishiwata
This paper deals with behavior of polygonal plane curves with two asymptotic lines by generalized crystalline curvature flow with a driving force. We show global existence of V-shaped solutions and investigate the deformation patterns of non-V-shaped solutions. We also show that non-V-shaped solutions become V-shaped in finite time.
keywords: Motion by crystalline curvature polygonal curves eventual monotonicity
DCDS-S
On spiral solutions to generalized crystalline motion with a rotating tip motion
Tetsuya Ishiwata
In our previous paper we proposed a crystalline motion of spiral-shaped polygonal curves with a tip motion as a simple model of a step motion on a crystal surface under screw dislocation and discussed global existence of spiral solutions to the proposed model. In this paper we extend the previous results for generalized crystalline curvature flow with a suitable tip motion. We show that solution curves never intersect a trajectory of a tip and has no self-intersections. We also show that any facet never disappear during time evolution. Finally we show a time-global existence of the spiral-shaped solutions.
keywords: Spirals crystalline curvature polygonal curves motion by curvature.

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