| 研究課題/領域番号 |
18K13440
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| 研究機関 | 金沢大学 |
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研究代表者 |
POZAR Norbert 金沢大学, 数物科学系, 准教授 (00646523)
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| 研究期間 (年度) |
2018-04-01 – 2022-03-31
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| キーワード | crystalline curvature / facet breaking / interacting particles / annihilation |
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| 研究実績の概要 |
We have succeeded in generalizing our notion of viscosity solutions for the crystalline mean curvature flow to problems with non-uniform driving force. It is a rather nontrivial generalization due to the complex interaction of the nonlocal crystalline mean curvature and the driving force, which can break or bend the crystal facets during the evolution. This is an important step to allow this problem to model a real crystal growth like the growth of snow crystals (snowflakes), where the conditions of the surrounding medium strongly vary depending on the position and time, leading to complex structures with breaking facets and a growth of dendrites.
We have been also exploring an interesting application of the viscosity solutions for a model of interacting particles with annihilation in one space dimension appearing in dislocation dynamics. The trajectories of particles can be expressed as contours of a so-called level set function that solves an appropriate nonlocal partial differential equation. The powerful tools of the viscosity theory allow for passing in the many particle limit to the continuum equation.
The manuscripts for the above results are in the final stage of preparation in collaboration with multiple researchers.
I have also analyzed an important explicit example of fattening (nonuniqueness) in the crystalline mean curvature flow in the context of our notion of viscosity solutions.
I have co-organized a mini-symposium "Singular parabolic equations and the motion of interfaces" at ICIAM2019 (Valencia, Spain).
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
We have made a significant progress on many of the planned points in the original proposal. Furthermore, by pursing a connection with annihilating particle systems, we are expanding on the possible applications of the viscosity solution method.
Unfortunately, the COVID-19 pandemic has forced me to cancel a planned trip to visit my collaborators. However, this did not significantly influence the progress since we proceeded with other research tasks and took advantage of online communication tools.
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| 今後の研究の推進方策 |
The near future will be spent by finishing the manuscripts for the new results outlined above. After that I will continue following the topics on the original proposal, with the focus on further generalization of the viscosity solutions for the crystalline mean curvature flow to other problems (non-purely crystalline anisotropies, inverse mean curvature flow), and the possibility of the analysis of facets in free boundary problems: the time dependent Hele-Shaw flow.
Due to the anticipated restrictions on travel in the near future, I will try to improve the research efficiency by using online-only communication.
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| 次年度使用額が生じた理由 |
Due to the travel restrictions related to the COVID-19 pandemic, I was forced to cancel a planned visit of my collaborators in the USA. My current plan is to postpone the visit until the travel restrictions are lifted. I will use a small part of the funds to acquire tools for better remote communication (like live handwritten communication important for math research: iPad Pro + Apple Pencil or similar)
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