Studies on mathematical structure of boundary value problems appearing in hydrodynamics and magnetohydrodynamics

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04957
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Nara Women's University
• Principal Investigator: YANAGISAWA Taku 奈良女子大学, 自然科学系, 教授 (30192389)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 境界値問題 / MHD方程式 / 安定性 / Hodge分解 / 調和ベクトル場

Outline of Final Research Achievements

This research was intended as an attempt to study the relations, from a mathematical standpoint, between the following three issues on the boundary value problems appearing in the hydrodynamics and magnetohydrodynamics (MHD) : (i) the nonlinear structure of equations, (ii) the setting of boundary conditions, (iii) the geometrical structure of domains. Our results obtained in this reserach can be listed below.
(1) We proved the existence and the stability of weak solutions of stationary MHD equations under inhomogeneous boundary conditions.(2) We gave a characterization of harmonic vector fields in three dimensional exterior domain by topological invariants of exterior domains. This result shall be a base for establishing Hodge decomposition theorem for exterior domains.

Free Research Field

非線型解析学