2001 Fiscal Year Final Research Report Summary
Nonlinear Analysis on diffusion effects producing singular shapes
| Project/Area Number |
10304010
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Hokkaido University |
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Principal Investigator |
GIGA Yoshikazu Hokkaido Univ., Grad. School of Sci., Prof., 大学院・理学研究科, 教授 (70144110)
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| Co-Investigator(Kenkyū-buntansha) |
OMATA Seirou Kanazawa Univ., Fac. of Sci., Asso. Prof., 理学部, 助教授 (20214223)
ITO Kazuo Kyushu Univ., Grad. School of Math. Sci., Asso. Prof., 大学院・数理学研究科, 助教授 (20280860)
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)
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| Project Period (FY) |
1998 – 2001
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| Keywords | nonlocal / higher order diffusion / facet / crystalline motion / level set method / anisotropy / Navier-Stokes equation / lower semicontinuity |
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| 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 Navier-Stokes 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 energy. 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 self-intersection 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 Navier-Stokes 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.
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