2022 Fiscal Year Annual Research 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 – 2023-03-31
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| Keywords | crystalline curvature / viscosity solutions / many particle limit / free boundary |
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| Outline of Annual Research Achievements |
We continued with the investigation of the viscosity solution method for important problems tracking the evolution of interfaces.
We published an extensive review paper that summarizes the recent developments on the crystalline mean curvature flow problem with the focus on its well-posedness: existence and uniqueness of generalized solutions. A large part is devoted to the results of this project. We included new proofs of various statements related to the crystalline mean curvature. We hope that this will serve as a reference for other researchers hoping to enter this field.
We also published a result on the many-particle limit in a simplified one-dimensional model of interacting dislocations in a crystalline lattice. Dislocation are defects of the lattice of crystals like metals that interact through elastic stress and their motion and annihilation influences plasticity properties of the material. Understanding of their behavior is an interesting and challenging mathematical problem. The dislocations can be viewed as level sets of a solution of a Hamilton-Jacobi equation and we were able to apply viscosity solution method to establish the continuum (many-particle) limit of this system.
Finally, we could restart a joint work on a free interface model of a contact line motion of a droplet on a rough surface with periodic structure that exhibits crystalline-like limit shapes. This problem features a combination of a variational structure as a rate-independent system and a viscosity solution structure, and we hope to apply the methods developed in this project.
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