2018 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 / Hele-Shaw problem / Stefan problem / self-similar solutions |
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| Outline of Annual Research Achievements |
We have made progress both in the study of crystalline mean curvature flow and Hele-Shaw type problems.
I have constructed explicit self-similar solutions of higher topological genus (shrinking objects with holes) for the crystalline mean curvature flow. Such solutions are useful to study the various singularities of the evolving crystal, including a pinch-off of a dumbbell shape. Furthermore, I have implemented a promising numerical method for the flow in three dimensions and tested its accuracy using the self-similar solutions. I hope to further develop this code to model the growth of realistic crystals of snow and other materials. I have obtained first results in this direction.
We have achieved further progress in understanding of the behavior of problems with a free boundary interacting with a highly oscillating medium, the so-called homogenization of Hele-Shaw and Stefan problems. With I. Palupi, we developed an efficient numerical method to find out the average velocity of the free boundary (water surface in a porous medium) when it is being perturbed by a space-time dependent irregularity. We have observed that the surface might develop flat parts resembling the facets of a crystal. This appears to be one of the first suggestions that a dynamic problem of this kind can exhibit facets stable in time. With G.T.T.Vu, we analyzed the behavior of an anisotropic model of phase transitions, the Stefan problem, in a highly oscillating medium in the large time limit.
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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 completed three papers concerning the research questions proposed in this project. The observation of appearance of flat parts of the fluid surface in an inhomogeneous porous medium in particular suggests a very interesting research direction to understand this phenomenon. At the same time, we have made progress on other aspects of the project. To name a few: (a) We are making good progress in generalizing the notion of viscosity solutions for the crystalline mean curvature flow to more general situations required by applications to material modeling like a space and time dependent forcing term. (b) We have a clearer understanding on when crystal facets break in the evolution, using a certain geometric minimization problems to characterize such situations.
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| Strategy for Future Research Activity |
I will continue the work according to the original proposal, focusing on the generalization of the crystalline mean curvature flow solutions, understanding the breaking of facets, showing the preservation of geometric properties in the curvature flow, as well as trying to understand the observed phenomenon of flat parts of a fluid surface in an inhomogeneous porous medium.
To help with achieving the goals, I plan to invite a few researchers for a short visit and a discussion of the topics. Furthermore, I am co-organizing a minisymposium at the ICIAM2019 conference in Valencia, Spain to bring together researchers working in closely related fields to exchange the newest ideas.
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| Causes of Carryover |
Due to scheduling difficulties, I have moved one of the planned international trips to the following fiscal year. This does not affect the execution of the goal of the project overall.
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