Existence of solutions of free boundary problems of two-phase fluids and their asymptotic behaviors

• Project/Area Number: 17K17804
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis; Mathematical analysis
• Research Institution: Nagoya University
• Principal Investigator: Terasawa Yutaka 名古屋大学, 多元数理科学研究科, 准教授 (90546160)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 二層流体 / 拡散界面モデル / ナビエ・ストークス方程式 / カーン・ヒリアード方程式 / 非局所作用素 / 局所漸近 / 弱解 / エネルギー不等式 / 二層流体問題 / 非局所自由エネルギー / 局所自由エネルギー / 定常ナヴィエ・ストークス方程式 / リュービル型定理 / 非局所エネルギー / ナヴィエ・ストークス方程式 / 漸近挙動 / 化学ポテンシャル / 領域分数ラプラシアン / Diffuse Interface Model / Phase-Field Model / 粘性流体 / 混相流 / 自由境界問題 / 実解析

Outline of Final Research Achievements

We investigated the existence of solutions for a two-phase fluid problem which describes behavior of a mixture of two fluids such as water and oil and relations of solutions for two kinds of such models. Diffuse interface model is a model where there is a thin mixed region of two fluids. The equations are described as a coupled system of the Navier-Stokes equations and the Cahn-Hilliard equations. We investigated the existence of solutions in a case where the Cahn-Hillard equations in the system has a nonlocal operator represented by some integral operator. We also investigated the relation between the solutions of nonlocal and local Navier-Stokes-Cahn-Hillard equations.

Academic Significance and Societal Importance of the Research Achievements

二層流体モデルは、水と油などの混合物の状態及び運動を説明する極めて重要なモデルである。その中でも拡散界面モデルは、流体塊がちぎれたり、くっついたりする場合の状況を記述できると期待され、工学的なシュミレーションの観点でもその有用性が高い。拡散界面モデルの非局所モデルという最近注目されている基本的なモデルについて、解の存在について調べ、また、非局所モデルの解と古典的な局所モデルの解の関係を調べた本研究は、拡散界面モデルの研究に一定の意義を有し、今後の研究の礎となると考えられる。

Research Products

• [Int'l Joint Research] 台湾師範大学(中国)
• [Int'l Joint Research] レーゲンスブルク大学(ドイツ)
• [Int'l Joint Research] レーゲンスブルク大学(ドイツ)
• [Int'l Joint Research] レーゲンスブルク大学(ドイツ)
• [Int'l Joint Research] レーゲンスブルク大学(ドイツ)
• [Int'l Joint Research] レーゲンスブルク大学(ドイツ)
• [Int'l Joint Research] ソウル国立大学(韓国)
• [Int'l Joint Research] レーゲンスブルク大学(ドイツ)
• [Int'l Joint Research] レーゲンスブルク大学(ドイツ)
• [Journal Article] Liouville-type theorems for the Taylor--Couette--Poiseuille flow of the stationary Navier--Stokes equations (2024)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Journal Title: Journal of Fluid Mechanics Volume: -
  • Peer Reviewed
• [Journal Article] Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition (2023)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Journal Title: Journal of Differential Equations Volume: 365 Pages: 905-926
  • DOI: 10.1016/j.jde.2023.05.025
  • Peer Reviewed / Open Access
• [Journal Article] Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities (2022)
  • Author(s): Helmut Abels and Yutaka Terasawa
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. S Volume: 15 Issue: 8 Pages: 1871-1871
  • DOI: 10.3934/dcdss.2022117
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral (2022)
  • Author(s): Hideo Kozono, Yutaka Terasawa and Yuta Wakasugi
  • Journal Title: Indiana Univ. Math. J. Volume: 71 Issue: 3 Pages: 1299-1316
  • DOI: 10.1512/iumj.2022.71.8978
  • Peer Reviewed
• [Journal Article] Asymptotic properties of steady solutions to the 3D axisymmetric Navier-Stokes equations with no swirl (2022)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Journal Title: Journal of Functional Analysis Volume: 282 Issue: 2 Pages: 109289-109289
  • DOI: 10.1016/j.jfa.2021.109289
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of solutions to elliptic and parabolic equations with unbounded coefficients of the second order in unbounded domains (2020)
  • Author(s): Hideo Kozono, Yutaka Terasawa and Yuta Wakasugi
  • Journal Title: Mahematische Annalen Volume: - Issue: 3-4 Pages: 1105-1117
  • DOI: 10.1007/s00208-020-02032-2
  • Peer Reviewed
• [Journal Article] Weak solutions for a diffuse interface model for two‐phase flows of incompressible fluids with different densities and nonlocal free energies (2020)
  • Author(s): Helmut Abels, Yutaka Terasawa
  • Journal Title: Mathematical Methods in the Applied Sciences Volume: Volume 43, Issue 6 Issue: 6 Pages: 3200-3219
  • DOI: 10.1002/mma.6111
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains (2018)
  • Author(s): H. Kozono, Y. Wakasugi, Y. Terasawa
  • Journal Title: J. Differential Equations Volume: 印刷中 Issue: 4 Pages: 1227-1247
  • DOI: 10.1016/j.jde.2018.03.027
  • Peer Reviewed / Open Access
• [Presentation] Liouville-type theorems for the Taylor-Couette flow of the stationary Navier-Stokes equations (2024)
  • Author(s): Yutaka Terasawa
  • Organizer: HMAセミナー・冬の研究会
  • Invited
• [Presentation] Liouville-type theorems for the Taylor-Couette flow of the stationary Navier-Stokes equations (2023)
  • Author(s): Yutaka Terasawa
  • Organizer: The 10th East Asian Conference in Harmonic Analysis and Applications
  • Int'l Joint Research / Invited
• [Presentation] Liouville-type theorems for the Taylor-Couette flow of the stationary Navier-Stokes equations (2023)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Organizer: 日本数学会2023年度秋季総合分科会
• [Presentation] Convergence of a Nonlocal to a Local Diffuse Interface Model for Two-Phase Flow with Unmatched Densities (2023)
  • Author(s): Yutaka Terasawa
  • Organizer: RIMS共同研究（公開型）発展方程式とその周辺--エネルギー構造と解の定量的解析--
  • Int'l Joint Research / Invited
• [Presentation] Convergence of a Nonlocal to a Local Diffuse Interface Model for Two-Phase Flow with Unmatched Densities (2023)
  • Author(s): Yutaka Terasawa
  • Organizer: Analysis Seminar, Ljubljana University
  • Int'l Joint Research / Invited
• [Presentation] Convergence of a Nonlocal to Local Diffuse Interface Nodel for Two-phase Flows with Unmatched Densities (2022)
  • Author(s): Yutaka Terasawa
  • Organizer: The 9th East Asian Conference in Harmonic Analysis and Applications
  • Int'l Joint Research / Invited
• [Presentation] 二層流体の非局所拡散界面モデルの弱解の存在及びその局所漸近 (2022)
  • Author(s): 寺澤祐高
  • Organizer: 日本数学会２０２２年度秋季総合分科会函数方程式論分科会特別講演
  • Invited
• [Presentation] Convergence of a Nonlocal to a Local Diffuse Interface Model for Two-Phase Flow with Unmatched Densities (2022)
  • Author(s): Yutaka Terasawa
  • Organizer: 第37回調和解析セミナー
• [Presentation] Asymptotic properties of steady solutions to the 3D axisymmetric Navier-Stokes equations with no swirl (2021)
  • Author(s): Yutaka Terasawa
  • Organizer: The 8th East Asian Conference in Harmonic Analysis and Applications (The 20th In- ternational Conference, Graduate School of Mathematics, Nagoya University)
  • Int'l Joint Research / Invited
• [Presentation] Weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies (2021)
  • Author(s): Yutaka Terasawa
  • Organizer: 第20 回非線形発展方程式セミナーat KUE
  • Invited
• [Presentation] Weak solutions for a diffuse interface model for two-phase ows of incompressible uids with different densities and nonlocal free energies (2021)
  • Author(s): Yutaka Terasawa
  • Organizer: SPRING 2021 Nonlinear Analysis Seminar Series, OIST (online on Zoom)
  • Int'l Joint Research / Invited
• [Presentation] Weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies (2020)
  • Author(s): Helmut Abels, Yutaka Terasawa
  • Organizer: Harmonic Analysis workshop in Seoul, Seoul National University, 2020年2月
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic properties of steady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet energy (2019)
  • Author(s): Hideo Kozono, Yuta Wakasugi, Yutaka Terasawa
  • Organizer: 7th East Asian Conference in Harmonic Analysis and Applications, Chung-Ang University, Seoul, Korea
  • Int'l Joint Research / Invited
• [Presentation] Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities and Nonlocal Free Energies (2019)
  • Author(s): Helmut Abels, Yutaka Terasawa
  • Organizer: 非線形解析セミナー、慶應義塾大学
  • Invited
• [Presentation] Existence of weak solutions fora diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies (2019)
  • Author(s): Helmut Abels and Yutaka Terasawa
  • Organizer: 第二十回北東数学解析研究会
  • Int'l Joint Research / Invited
• [Presentation] Existence of weak solutions fora diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies (2019)
  • Author(s): Helmut Abels and Yutaka Terasawa
  • Organizer: 日本数学会年会, 東京工業大学
• [Presentation] Asymptotic properties of steady solutions to the 2D Navier-Stokes equations with Finite Generalized Dirichlet Integral (2019)
  • Author(s): 小薗英雄, 寺澤祐高, 若杉勇太
  • Organizer: 日本数学会年会, 東京工業大学
• [Presentation] Weak Solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies (2018)
  • Author(s): Helmut Abels and Yutaka Terasawa
  • Organizer: International Conference on Harmonic Analysis and Applications, Yanqi Lake, Beijing
  • Int'l Joint Research / Invited
• [Presentation] Weak Solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies (2018)
  • Author(s): Helmut Abels and Yutaka Terasawa
  • Organizer: The 6th East Asian Conference in Harmonic Analysis and Applications, Osaka University, Toyonaka
  • Int'l Joint Research
• [Presentation] Weak Solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies (2018)
  • Author(s): Helmut Abels and Yutaka Terasawa
  • Organizer: Mathematical Analysis of Viscous Incompressible Fluids, RIMS, Kyoto
  • Int'l Joint Research / Invited
• [Presentation] A remark on Liouville-type theorem for the nonstationary Navier-Stokes equations in two dimensional domains (2018)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Organizer: 日本数学会年会, 東京大学
• [Presentation] M. T. Lacey著 "An elementary proof of A2 bound", Israel J. Math. 217 (2017), no.1, p.181-p.195の紹介 (2018)
  • Author(s): Yutaka Terasawa
  • Organizer: Sparse bound 勉強会
• [Presentation] Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains (2017)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Organizer: 5th East Asain Conference in Harmonic Analysis and its Applications, Zhejiang, China
  • Int'l Joint Research / Invited
• [Presentation] Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains (2017)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Organizer: Harmonic Analysis and its applications in Tokyo 2017, Tokyo, Japan
  • Int'l Joint Research / Invited
• [Presentation] Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains (2017)
  • Author(s): Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi
  • Organizer: 日本数学会秋季総合分科会, 山形大学
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• [Funded Workshop] 調和解析と非線形偏微分方程式 (2022)
• [Funded Workshop] The 8th East Asian Conference in Harmonic Analysis and Applications (2021)