2005 Fiscal Year Final Research Report Summary
Mathematical Analysis on Change of Patterns by Anisotropy and Diffusion
| Project/Area Number |
14204011
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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 | University of Tokyo (2004-2005)
Hokkaido University (2002-2003) |
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Principal Investigator |
GIGA Yoshikazu University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70144110)
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| Co-Investigator(Kenkyū-buntansha) |
ARAI Hitoshi University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (10175953)
TONEGAWA Yoshihiro Hokkaido University, Department of Mathematics, Assistant Professor, 大学院・理学研究科, 助教授 (80296748)
FURUKAWA Yoshinori Hokkaido University, Institute of Low Temperature Science, Assistant Professor, 低温科学研究所, 助教授 (20113623)
NAKAJI Takahiko Hokkaido University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (30002174)
TSUTAYA Kimitoshi Hokkaido University, Department of Mathematics, Assistant Professor, 大学院・理学研究科, 助教授 (60250411)
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| Project Period (FY) |
2002 – 2005
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| Keywords | anisotropy / Berg's effect / selfsimilar solution / proper viscosity solution / nonlinear diffusion / singular vertical diffusivity / Burgers equation / Navier-Stokes equation |
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| Research Abstract |
Several mathematical foundation is established to investigate the mechanism of variation of shape of solutions due to anisotropy and diffusion effect for nonlinear diffusion equations. We just give two typical topics.
(i)Equation describing motion of a crystal surface. When anisotropy of surface energy is strong like snow crystal, there appears a plat surface called a facet in their equilibrium shape. The equations governing its evolution is formally a curvature flow equation. However, the diffusion is so strong that it cannot be viewed as a partial differential equation. We studied a Stefan type free boundary problem describing snow crystal growth assuming that its equilibrium shape is a cylinder. We established (a)Local solvability under the assumption that a facet stays as a facet ; (b)Berg's effect concerning behavior diffusion field near on a facet ; (c)a necessary and sufficient condition for facet bending ; (d)a sufficient condition for existence of a self-similar solution and size condition on facet bending ; (e)a rigorous proof that a facet preserved near equilibrium. A further problem is whether one can track evolution after facet bending.
(ii)Equations in fluid mechanics : An invicid Burgers equation is a typical coarse model to describe motion of compressible fluid. The solution develops jump discontinuity in finite time even if the initial data is smooth for the Burgers equation. Just before this proposal, the principal inverstigator introduced a notion of a proper viscosity solution to describe a solution with jumps which is also useful for equations with nondivergence type. In this project we established a notion of singular vertical diffusivity which is useful to calculate the solution numerically in the graph space. In particular we are successful to calculate a proper viscosity solution by a level set method numerically.
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