Mathematical analysis of self-similar structure for nonlinear partial differential equations

2017 Fiscal Year Final Research Report

• Project/Area Number: 25707005
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Partial Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: Miura Hideyuki 東京工業大学, 情報理工学院, 准教授 (20431497)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: 自己相似解 / 調和写像熱流 / 非圧縮性流体

Outline of Final Research Achievements

The harmonic map heat flow equation and partial differential equations describing the incompressible fluids are studied. Concerning the harmonic map heat flow, we focused on equivariant maps from the n-dimensional Euclidean space to the n-dimensional sphere in energy supercritical dimensions and showed that certain data can give rise to two distinct solutions which are both stable. We also identified the optimal condition for the range of the solution to guarantee the uniqueness. In the research of the incompressible fluids, various results such as the Helmholtz decomposition and the structure theorem for the incompressible vector fields in unbounded domains are obtained.

Free Research Field

偏微分方程式論