Well-posedness for the nonlinear partial differential equations in critical spaces

2022 Fiscal Year Final Research Report

• Project/Area Number: 17H04824
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Mathematical analysis
• Research Institution: Tohoku University
• Principal Investigator: IWABUCHI Tsukasa 東北大学, 理学研究科, 准教授 (40634697)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: Navier-Stokes方程式 / スペクトル理論

Outline of Final Research Achievements

We obtained several results for compressible Navier-Stokes equations under the barotropic condition and for the full-system of the density, the velocity and the temperature. The ill-posedness is proved in the function spaces with the threshold regularity for each equations. In addition, we proved the existence of self-similar solutions under the radial symmetric condition for the density, the velocity and the temperature.
As for theory of the function spaces on general domains and its application to nonlinear partial differential equations, we established one technique with the dyadic decomposition of the spectrum for the Dirichlet Laplacian applicable to the surface quasi-geostrophic equation in the half space and the bounded domain.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

非線形偏微分方程式の初期値問題の適切性および非適切性の問題は線形と非線形の性質の釣り合いの状況を数学的に考察する問題である。本研究ではこうした問題を流体力学に関連する方程式に対して行っており、適切性のための臨界空間を明らかにすることで解の安定性を判定する際の一定の方法を提示したことになると考える。領域上の解析理論に関しては、数学的に解の一意性や安定性を議論するための基礎理論の構築の一部である。