Real analytic research on the Navier-Stokes equations on exterior domains with external force

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05339
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Waseda University
• Principal Investigator: Yamazaki Masao 早稲田大学, 理工学術院, 教授 (20174659)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: Navier-Stokes方程式 / 外部領域 / 安定性 / Lorentz空間 / 実補間

Outline of Final Research Achievements

The author investigated Navier-Stokes equations, which describe motion of incompressible fluids, in 2-dimensional unbouded domains. (In particular exterior Gdomains.)
The author first considered stationary problem in exterior domains. No genaral condition was obtained for the existence of solutions which decay at infinity. The author showed the unique existence of weak solutions satisfying energy inequality under the assumption under newly introduced symmetry assumptions on domains, external forces and the boundary conditions.
The author next considered on the whole plane in the case where external forces depends on time. Under the aforementioned symmetry condition on the external forces, the author showed the unique existence and stability of small solutions. in oreder to overcome the weak decay in 2-dimension, we employed the weighted Lorentz spaces.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

Navier-Stokes 方程式は航空機、船舶、鉄道車両、自動車盗の製造に重要な役割を果たす。本研究は具体的な問題に直接かかわっていないが、Navier-Stokes方程式に新たな知見を与え, その研究の進展にきよするものである。