Development of Viscosity and Variational Techniques for the Analysis of Moving Interfaces

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13440
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kanazawa University
• Principal Investigator: Pozar Norbert 金沢大学, 数物科学系, 准教授 (00646523)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: generized solutions / crystal growth / mean curvature / annihilation / comparison principle

Outline of Final Research Achievements

Interfaces that separate different moving regions are a feature of many important problems in physics, material science and biology. The mathematical understanding of equations arising in such problems is crucial for developing accurate numerical methods to find a solution on a computer. The interfaces might collide or split, and a proper continuation of solution past such times is mathematically challenging. We contributed to the theory of generalized solutions of a model of crystal growth, showing existence and uniqueness of a new notion of solution of the so called crystalline mean curvature flow with space dependent forcing, which will allow the use of this model in more realistic situation like the growth of snow crystals. We also studied a related problem with a prescribed volume of the crystal. We used similar mathematical tools to study how a large number of dislocations in a material move. Their movement governs how the material will respond to deformations.

Free Research Field

関数方程式

Academic Significance and Societal Importance of the Research Achievements

研究した数学問題は、物理学、材料科学、生物学、人口動態などのさまざまな分野の重要なモデルに使われます。得られた理論的結果は、これらの問題の特徴についての理解を深め、コンピューター上で解を求めるための精度が高い数値解法を開発するための重要な基盤となります。これらのモデルを結晶成長、材料の塑性、土壌中の水の流れなどの応用研究に活用していただければ幸いです。