Large time behavior of solutions to partial differential equations with supercritical nonlinearity

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05312
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: Miura Hideyuki 東京工業大学, 理学院, 教授 (20431497)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 非線形偏微分方程式

Outline of Final Research Achievements

We studied the initial-boundary value problem and regularity of the solutions to
the incompressible Navier-Stokes equations. In particular, it was shown that there exist global in time weak solutions for the initial-boundary problem for the Navier-Stokes equations in the three dimensional half space for initial data with infinite energy. As for regularity of the solutions, we showed that if a scaled energy of the initial velocity is sufficiently small in some neighborhood, the weak solution is locally smooth at least for a short time. Furthermore, we obtained new estimates on regions where the weak solutions are smooth for initial data in the weighted spaces of square integrable functions.

Free Research Field

非線形偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

非圧縮性Navier-Stokes方程式は流体力学の基礎方程式としての重要性から古くから多くの研究が行われている．しかし，方程式のもつ非局所性から生じる困難から，エネルギーが無限大となるような特異性をもつ速度場の研究は未解明の部分が多かった．本研究では非局所性を扱う上で鍵となる圧力の評価に関して新しい技術を導入することにより，弱解の時間大域的存在や局所正則性に関する成果を得ることができた．今回用いられた手法は非圧縮性粘性流体の数理解析における今後の研究においても有用になると期待できる．