Mathematical analysis of anisotropy and singular limit problems in the equations of geophysical fluid dynamics

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03584
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: The University of Tokyo (2022); Kyushu University (2019-2021)
• Principal Investigator: Takada Ryo 東京大学, 大学院数理科学研究科, 准教授 (50713236)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 偏微分方程式 / Navier-Stokes方程式 / 磁気流体力学方程式 / Coriolis力 / 時間大域的適切性 / 特異極限問題 / 長時間挙動

Outline of Final Research Achievements

The aim of this research is the mathematical analysis for nonlinear partial differential equations arising in geophysical fluid dynamics and magnetohydrodynamics. We consider the initial value problems for the incompressible Navier-Stokes equations and the magnetohydrodynamics equations with the Coriolis force, and show the global well-posedness for the initial data in the scaling critical Sobolev spaces provided that the rotating speed is sufficiently high. Also, we investigate the large time behavior of global solutions to the rotating Navier-Stokes equations, and derive the temporal decay estimates and the asymptotics of solutions as time approaches infinity.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

流体力学に現れる非線形偏微分方程式系の数学解析において，初期値問題に対する時間大域解の存在と一意性，および解の長時間挙動の解析は基礎的かつ重要な研究課題である．本研究では，回転の影響による Coriolis 力付き非圧縮性 Navier-Stokes 方程式および磁気流体方程式を対象として上記の問題に取り組み研究成果を得た．特に，回転の影響による流れの長時間挙動の変化を，時間減衰評価および漸近形の観点から特徴付けることに成功した．