2015 Fiscal Year Annual Research Report
変分法を用いた液適や泡の運動モデルに対する数理解析
| Project/Area Number |
25800087
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| Research Institution | Hokkaido University |
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Principal Investigator |
Ginder Elliott 北海道大学, 電子科学研究所, 助教 (30648217)
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| Project Period (FY) |
2013-04-01 – 2016-03-31
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| Keywords | droplet motion / threshold dynamics / free boundary problems / variational methods / numerical algorithm |
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| Outline of Annual Research Achievements |
We developed an analytic framework for approximating solutions to a model equation whose evolutions express motions related to droplets and bubbles. Our approach successfully utilized the method of minimizing movements (MM) to treat the phenomena using energy theoretic techniques. By examining free boundary conditions, we incorporated droplet contact angle dynamics into the variational formulation of our problems and developed methods for investigating numerical properties. Using our methods we simulated droplet dynamics (including their coalesce and divisions). We also discovered and analyzed a threshold dynamical algorithm for approximating motion by hyperbolic mean curvature flow and derived method’s order of convergence. Moreover, through the combination of mathematical and computational analysis, we have clarified essential properties of our equations' variational structures. This was achieved by establishing the framework for employing MMs and has enabled us to advance both the simulations and the mathematical analysis. In particular, we were able to reach a major goal of this research, which was to permit that the analysis of the model equation to be performed in the setting of MMs, both mathematically and computationally.
New questions have arisen from this research and we want to continue our analysis to understand their answers. Also, we are excited that the techniques generated in this research can be used in analyzing new problems.
This research was recognized the Mathematical Society of Japan and received the “MSJ Prize for Excellent Applied Mathematicians.”
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