Project/Area Number  10304010 
Research Category 
GrantinAid for Scientific Research (A).

Section  一般 
Research Field 
Global analysis

Research Institution  Hokkaido University 
Principal Investigator 
GIGA Yoshikazu Hokkaido Univ., Grad. School of Sci., Prof., 大学院・理学研究科, 教授 (70144110)

CoInvestigator(Kenkyūbuntansha) 
OMATA Seirou Kanazawa Univ., Fac. of Sci., Asso. Prof., 理学部, 助教授 (20214223)
JIMBO Shuichi Hokkaido Univ., Grad. School of Sci., Prof., 大学院・理学研究科, 教授 (80201565)
SATO Motohiko Muroran Inst. Tech., Fac. of Eng., Asso. Prof., 工学部, 助教授 (30254139)
KOBAYASHI Ryo Hokkaido Univ., Research Inst. Elect. Sci., Asso. Prof., 電子科学研究所, 助教授 (60153657)
ITO Kazuo Kyushu Univ., Grad. School of Math. Sci., Asso. Prof., 大学院・数理学研究科, 助教授 (20280860)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥38,830,000 (Direct Cost : ¥36,100,000、Indirect Cost : ¥2,730,000)
Fiscal Year 2001 : ¥11,830,000 (Direct Cost : ¥9,100,000、Indirect Cost : ¥2,730,000)
Fiscal Year 2000 : ¥7,800,000 (Direct Cost : ¥7,800,000)
Fiscal Year 1999 : ¥7,500,000 (Direct Cost : ¥7,500,000)
Fiscal Year 1998 : ¥11,700,000 (Direct Cost : ¥11,700,000)

Keywords  nonlocal / higher order diffusion / facet / crystalline motion / level set method / anisotropy / NavierStokes equation / lower semicontinuity / 非局所的曲率 / 高階拡散 / ファセット / クリスタライン運動 / 等高面法 / 異形性 / ラヴィエ・ストークス方程式 / 下半連続性 / 粘性解 / 非局所的拡散効果 / 適正粘性解 / 異方性 / ナヴィエ・ストークス方程式 / 2次元流 / 界面支配モデル / 表面エネルギー / 等高面の方法 / 非局所的曲率流方程式 / 非線形拡散方程式 / 非線形 / 非平衡 / 拡酸 / 高階微分方程式 
Research Abstract 
Diffusion effects often play a key role in nonlinear nonequilibrium phenomena. To understand its role we mainly studied the following topics (i) interfacial dynamics with nonlocal curvature (ii) interface dynamics with higher order diffusion (iii) fold energy (iv) the NavierStokes initial value problem. (i) Evolution of crystal surface is described by a curvature flow equation with a driving force in the interface controlled models. There are several situations that anisotropy of interfacial energy is so strong that its Wulff shape has a flat portion called a facet. In this case even the notion of solution was unclear since the diffusion effect is considered nonlocal. For an evolving curve we introduce a notion of solutions by extending a level set method and prove the comparison and convergence theorems. As applications of our results we prove the convergence of crystalline algorithm for general equations and that the crystalline motion is a limit of the motion by smooth interfacial e
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nergy. These works are highly nontrivial since curvature effect is nonlocal. To complete the theory we published more than 200 pages. (ii) Motion by surface diffusion includes a fourth order diffusion effect. Compared with second order model, the behavior of solution is quite different. For example, we proved that a solution may loose convexity as well as non selfintersection properly in a finite time. (iii) This energy is considered as a limit of energy whose density include the second order derivatives. We have proved its lower semicontinuity which triggers a lot of other works in this direction. (iv) We have proved the unique existence of smooth global planer NavierStokes flow even when initial data is merely bounded, many not decay at space infinity. Besides these works we have published a book "Nonlinear Partial Differential Equation" (Kyoritsu) including topics on pinching phenomena of mean flow and large time behavior of solutions of vorticity equations with several new results. この研究のほかに'非線形偏微分方程式'(共立)という単行本を出版した。この本では、平均曲率流方程式のちぎれ現象や過度方程式の解の時間大での挙動について新しい結果をも含めて解説した。 Less
