Qualitative and quantitative analysis of non-periodic space-time homogenization problems for nonlinear diffusion equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 22K20331
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: Oka Tomoyuki 東京大学, 大学院工学系研究科(工学部), 特任助教 (00962268)
• Project Period (FY): 2022-08-31 – 2024-03-31
• Keywords: 時空均質化問題 / 非線形拡散方程式 / 時空unfolding法 / 時空2スケール収束

Outline of Final Research Achievements

This study derived the homogenized equation for the space-time homogenization problem of nonlinear diffusion equations. Furthermore, the homogenized matrix was characterized, and it was clarified that differences in nonlinear diffusion deeply depend on the representation of the homogenized matrix. In particular, it was proved that the gradient of the solution does not converge strongly without imposing additional assumptions on the coefficient matrix field. Besides, H-convergence for a nonlinear problem was proved.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

均質化問題は，数学のみならず材料科学を始めとした諸分野と深く関係しており，最適設計問題への応用の観点から，均質化問題の研究が進展することは学術的のみならず社会的にも重要である．また，既存の静的線形問題に対する均質化理論を動的非線形問題へと拡張することは，数学的理論の深化に留まらず，新たな融合領域研究の基礎も成す．