2019 Fiscal Year Research-status Report
Development of Viscosity and Variational Techniques for the Analysis of Moving Interfaces
| Project/Area Number |
18K13440
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| Research Institution | Kanazawa University |
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Principal Investigator |
POZAR Norbert 金沢大学, 数物科学系, 准教授 (00646523)
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Keywords | crystalline curvature / facet breaking / interacting particles / annihilation |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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| Causes of Carryover |
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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