Mathematical Analysis of the incompressible viscous fluid with the moving contact line

2021 Fiscal Year Final Research Report

• Project/Area Number: 20K22311
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Waseda University
• Principal Investigator: Watanabe Keiichi 早稲田大学, 理工学術院, 講師(任期付) (30875365)
• Project Period (FY): 2020-09-11 – 2022-03-31
• Keywords: ナビエ・ストークス方程式 / 自由境界問題 / 最大正則性 / 関数方程式論 / 接触角

Outline of Final Research Achievements

The free boundary problem of the Navier-Stokes equations is considered with the 90 degrees contact angle condition. Here, the fluid occupies a three-dimensional bounded domain, and free boundary conditions and slip boundary conditions are imposed on the boundary of the domain. In this study, for a given time, the local appropriateness of the system is proved with Lp-in-time and Lq-in-space framework.

The stability of the free boundary problem of the Navier-Stokes equations is also studied, where the domain occupied by the fluid is bounded surrounded by a smooth boundary. The stability of the axisymmetric stationary solution is characterized by the positiveness of the second variation of the energy functional associated with the Euler-Lagrangian equation which determines the free boundary in the equilibrium.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

接触角を伴う Navier-Stokes 方程式の自由境界問題の適切性に関する本研究の結果は，Wilke (2013) で考えられていた境界条件の一部を修正し，より一般の関数空間で方程式系の適切性を示したものである．
一方，表面張力を伴う Navier-Stokes 方程式の自由境界問題の（軸対称な）非自明な定常解の安定性についての特徴づけの結果は，1800年代の Plateau の古典的結果を正当化する大変興味深いものである.