2016 Fiscal Year Research-status Report
Viscosity methods in homogenization of nonlinear PDEs
| Project/Area Number |
26800068
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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, 助教 (00646523)
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | homogenization / crystal growth / viscosity solutions / Stefan problem / phase transitions |
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| Outline of Annual Research Achievements |
Various contributions to the mathematical understanding of models of crystal growth, cancer, melting of solids and others have been achieved.
With Y. Giga (U of Tokyo), we have completed the development of a notion of viscosity solutions (generalized solutions) for models of a crystal growth driven by the surface energy with purely crystalline anisotropy. This provides a rigorous mathematical framework for applying the popular level set method to the study of evolving crystals with flat facets on the surface in three dimensions (and higher).
A robust numerical algorithm to compute the evolution of crystals studied in the above work has been implemented in two and three dimensions using the level set formulation of the problem, based on the idea of so-called minimizing movements. This algorithm was further combined with a model of heat flow outside of the crystal to give a physically more accurate description of the growth of real crystals, for instance snow flakes, in two dimensions. The results were presented, among others, in an invited lecture at the Spring meeting of Math. Society of Japan.
With G. T. T. Vu (Kanazawa U), we studied the behavior of the Stefan problem for large times. The Stefan problem models phase transitions like ice melting. We consider the case of varying properties of ice, for instance due to randomly located air bubbles inside. We quantify how the varying properties of ice influence the average melting speed. We use a combination of viscosity and variational techniques of the homogenization theory, partly developed in this project.
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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
As mentioned already in the summary above, we have finished important results as envisioned in this project. Even thought the current research direction is at this moment slightly departing from the original proposal, as was anticipated, the work on this project generated new very interesting topics to pursue.
For instance, we can now attempt to rigorously justify the numerical method for solving the crystal growth model in three dimensions using the developed notion of viscosity solutions, and this can further guide its development.
Similarly, with I. Kim (UCLA) we are now finishing a paper on the singular limit approximation of a model of mass transport due to a given drift field with a congestion constraint by models with "soft" constraints. This result has application to models of tumor growth in cancer modeling, crowding models, etc. Viscosity solutions techniques, in part developed in the current project, have been important in this work.
We have been also able to involve doctoral students.
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| Strategy for Future Research Activity |
The last year of the project will be spend on further exploring the viscosity solution techniques for free boundary problems and finishing the ongoing work.
Some time will be spend on finishing the paper with I. Kim (UCLA) of singular limit approximation of constrained mass transport models, see above in Reasons section. We can currently handle a drift field that concentrates the material. We will further investigate the possibility to extend our approach to general drift fields, which seems rather challenging.
The implemented numerical algorithm for the crystal growth model will be further improved. I plan to combine the surface energy driven model with appropriate heat transfer or diffusion model outside of the crystal in three dimensions. I hope to observe the formation of realistic snow crystals. It will be very interesting to investigate the influence of inhomogeneities on the growth. This will hopefully generate further ideas for future theoretic research.
With G. T. T. Vu (Kanazawa U.), we are currently working on extending our Stefan problem asymptotics result above to general nonanisotropic heat conductivities in the melted material. This will help us understand the interaction of the inhomogeneous diffusion of heat and the inhomogeneity of the melting material in long times.
I will continue to work on the homogenization of free boundary problems with nonmonotone evolution. Lastly, I plan to attend various meetings throughout the year to popularize the results of this project.
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| Causes of Carryover |
The cost of the planned trips during the previous year has been slightly lower than expected.
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| Expenditure Plan for Carryover Budget |
The transferred amount from last year will be used together with the budget for this year towards research visits, attendance of various meeting throughout the year to present result of this project as well as to invite collaborators and other researchers for discussions.
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