Mathematical analysis of two-phase flow equations in unbounded domains

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K14224
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: The University of Electro-Communications (2019-2020); Tokyo University of Science (2018); Waseda University (2017)
• Principal Investigator: Saito Hirokazu 電気通信大学, 情報理工学域, 准教授 (30754882)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 二相流 / 粘性流体 / 非有界領域 / 最大正則性 / 時間減衰評価 / Helmholtz分解 / 二相Navier-Stokes方程式 / Navier-Stokes-Korteweg

Outline of Final Research Achievements

The aim of this research project is the mathematical analysis for two-phase flow equations in unbounded domains. First, we proved the local wellposedness of two-phase Navier-Stokes equations with a sharp interface in general domains. In addition, we considered the two-phase Navier-Stokes equations in the whole space. We then established the global wellposedness for small initial data and a closed interface, and also proved time-decay estimates of a time evolution operator for a nearly flat interface. Next, we considered the Navier-Stokes-Korteweg equations which are known as one of diffuse interface models and proved the maximal regularity theorem for the linearized problem. As an application of the maximal regularity theorem, we proved the wellposedness of the Navier-Stokes-Korteweg equations.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

二相流基礎方程式として，シャープな界面を伴う二相Navier-Stokes方程式およびNavier-Stokes-Korteweg方程式を非有界領域において考察した．非有界領域における二相流の場合には，Helmholtz分解や最大正則性といった数学解析のための基礎的な理論が構築されていなかった．本研究では，そのような基礎的な理論の構築からはじめて，二相流基礎方程式の時間局所的適切性や時間大域的適切性を証明し，線形化問題および非線形問題の解の長時間挙動を明らかにした．