Construction of Mathematical Theory of 2 phase fluid flows

2020 Fiscal Year Annual Research Report

• Project/Area Number: 17H01097
• Research Institution: Waseda University
• Principal Investigator: 柴田 良弘 早稲田大学, 理工学術院, 教授 (50114088)
• Co-Investigator(Kenkyū-buntansha): 吉村 浩明 早稲田大学, 理工学術院, 教授 (40247234) ; 舟木 直久 早稲田大学, 理工学術院, 特任教授 (60112174) ; 小澤 徹 早稲田大学, 理工学術院, 教授 (70204196)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 流体方程式 / 2相問題 / 変分構造 / 確率偏微分方程式 / R-有界作用素 / 数値シミュレーション

Outline of Annual Research Achievements

1)マクロレベルからの流体研究
レゾルベント問題に対するR-有界作用素の理論をその流体数学への応用に関する研究を行い初期値境界値問題の最大正則性原理の確立に統一的な方法を与えた.これにより流体数学に表れる非線形問題の時間局所解の一意存在の証明は一般論をもって確立される.一方初期値・境界値問題の時間大域解、時間周期解の存在は低周波部分の解析が本質となるがこれについても多くの新しい知見を得た.具体的な成果としては, 多成分流体運動 MHD方程式の2相問題, 気液混合流体の運動を記述するNavier-Stokes-Korteweg方程式系 Liquid CrestalのQ-tensorモデルについてR-有界作用素の枠組みで考察し時間局所解の一意存在を示した.これらの問題に対し, Lp-Lq減衰定理を示し時間大域解の一意存在と漸近挙動を示した.
2)ミクロレベルからの流体研究
ランダム環境内の相互作用粒子系の極限として得られる放物型非線形確率偏微分方程式について, パラコントロール解析に基づき解の定式化を行い時間大域解の存在および定常解への収束などを示した.非平衡熱流体力学系の数学的な定式化と非定常挙動の解析に関する研究を行い非平衡開放形におけるディラック構造を示した.また定常的な2相流に関する不可逆過程の熱力学系についてエントロピー最小生成の原理による変分的定式化を行った.キャビテーションクラウドの2次元非定常流れに関しクラウドの生成から成長, 圧壊と流れ場の関連を数値解析的に解明した.

Current Status of Research Progress

1: Research has progressed more than it was originally planned.

Reason

当初の計画では扱う問題はNavier-Stokes方程式と相転移を考慮したNavier-Stokes-Korteweg方程式に研究を限っていたが,研究方法に注目され海外研究者に知己を得, 数学的厳密な扱いがなされていなかった, 多層流体, Liquid CrestalのQ-tensorモデルがR-有界性理論を用いできる事が分かり, この理論の有効性をさらに広げることができたから.

Strategy for Future Research Activity

1)マクロからの研究
これまでの研究をさらに圧縮性粘性流体の研究に広げるとともに, 周期解の関しR-有界作用素の観点からの研究を行う.
2)ミクロからの研究
これまでの研究を続けるとともに, ランダム環境内の相互作用粒子系の極限として自然に得られる特異な準線形放物型確率偏微分方程式の時間局所解の一意存在を示す.
またレイリー・ベナール対流の流体混合に関するモデルの周期解とその分岐現象の構造を明らかにする.

Research Products

• [Journal Article] Poincar? inequalities with exact missing terms on homogeneous groups (2021)
  • Author(s): OZAWA Tohru、SURAGAN Durvudkhan
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 73 Pages: 497～503
  • DOI: 10.2969/jmsj/83738373
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On the maximal L-L regularity of solutions to a general linear parabolic system (2020)
  • Author(s): Piasecki Tomasz、Shibata Yoshihiro、Zatorska Ewelina
  • Journal Title: Journal of Differential Equations Volume: 268 Pages: 3332～3369
  • DOI: 10.1016/j.jde.2019.09.058
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On the maximal Lp-Lq regularity theorem for the linearized electro-magnetic field equaitons with interface conditions (2020)
  • Author(s): E. Frolova and Y. Shibata
  • Journal Title: Zapiski Nauchn. Sem. POMI Volume: 489 Pages: 130～172
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On periodic solutions for one-phase and two-phase problems of the Navier?Stokes equations (2020)
  • Author(s): Eiter Thomas、Kyed Mads、Shibata Yoshihiro
  • Journal Title: Journal of Evolution Equations Volume: 21 Pages: 2955～3014
  • DOI: 10.1007/s00028-020-00619-5
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Some Free Boundary Problem for Two-Phase Inhomogeneous Incompressible Flows (2020)
  • Author(s): Saito Hirokazu、Shibata Yoshihiro、Zhang Xin
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 52 Pages: 3397～3443
  • DOI: 10.1137/18m1225239
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] The Global Well-Posedness for the Compressible Fluid Model of Korteweg Type (2020)
  • Author(s): Murata Miho、Shibata Yoshihiro
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 52 Pages: 6313～6337
  • DOI: 10.1137/19M1282076
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Existence and uniqueness of classical paths under quadratic potentials (2020)
  • Author(s): Narita Kazuki、Ozawa Tohru
  • Journal Title: Calculus of Variations and Partial Differential Equations Volume: 59 Pages: 128
  • DOI: 10.1007/s00526-020-01791-9
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Global solutions to the Maxwell?Navier?Stokes system in a bounded domain in 2D (2020)
  • Author(s): Fan Jishan、Ozawa Tohru
  • Journal Title: Zeitschrift f?r angewandte Mathematik und Physik Volume: 71 Pages: 136
  • DOI: 10.1007/s00033-020-01364-y
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Sharp remainder of the Poincar? inequality (2020)
  • Author(s): Ozawa Tohru、Suragan Durvudkhan
  • Journal Title: Proceedings of the American Mathematical Society Volume: 148 Pages: 4235～4239
  • DOI: 10.1090/proc/15119
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Multidimensional inverse Cauchy problems for evolution equations (2020)
  • Author(s): Karazym Mukhtar、Ozawa Tohru、Suragan Durvudkhan
  • Journal Title: Inverse Problems in Science and Engineering Volume: 28 Pages: 1582～1590
  • DOI: 10.1080/17415977.2020.1739034
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Well-posedness of a 2D time-dependent Ginzburg-Landau superconductivity model (2020)
  • Author(s): Fan Jishan、Ozawa Tohru
  • Journal Title: Nonlinear Analysis and Differential Equations Volume: 8 Pages: 89～97
  • DOI: 10.12988/nade.2020.91126
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion (2020)
  • Author(s): Funaki Tadahisa、Gao Yueyuan、Hilhorst Danielle
  • Journal Title: Mathematical Methods in the Applied Sciences Volume: 43 Pages: 5860～5886
  • DOI: 10.1002/mma.6329
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Dirac structures and variational formulation of port-Dirac systems in nonequilibrium thermodynamics (2020)
  • Author(s): Gay-Balmaz Francois、Yoshimura Hiroaki
  • Journal Title: IMA Journal of Mathematical Control and Information Volume: 37 Pages: 1298～1347
  • DOI: 10.1093/imamci/dnaa015
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Dirac structures in nonequilibrium thermodynamics for simple open systems (2020)
  • Author(s): Gay-Balmaz Francois、Yoshimura Hiroaki
  • Journal Title: Journal of Mathematical Physics Volume: 61 Pages: 092701～092701
  • DOI: 10.1063/1.5120390
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] From variational to bracket formulations in nonequilibrium thermodynamics of simple systems (2020)
  • Author(s): Gay-Balmaz Francois、Yoshimura Hiroaki
  • Journal Title: Journal of Geometry and Physics Volume: 158 Pages: 103812～103812
  • DOI: 10.1016/j.geomphys.2020.103812
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Semitheoretical formulation of annular flow void fraction using the principle of minimum entropy production (2020)
  • Author(s): Giannetti Niccol?、Yamaguchi Seiichi、Saito Kiyoshi、Yoshimura Hiroaki
  • Journal Title: International Journal of Thermal Sciences Volume: 158 Pages: 106522～106522
  • DOI: 10.1016/j.ijthermalsci.2020.106522
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Formulation of steady-state void fraction through the principle of minimum entropy production (2020)
  • Author(s): GIANNETTI Niccolo、SAITO Kiyoshi、YOSHIMURA Hiroaki
  • Journal Title: Journal of Thermal Science and Technology Volume: 15 Pages: JTST0025
  • DOI: 10.1299/jtst.2020jtst0025
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] R-bounded solution operators and mathematical fluid dynamics (2021)
  • Author(s): Yoshihiro Shibata
  • Organizer: Fudan International Seminar on Analysis, PDEs, and Fluid Mechanics
  • Int'l Joint Research / Invited
• [Presentation] R-bounded solution operators and mathematical fluid dynamics (2021)
  • Author(s): Yoshihiro Shibata
  • Organizer: A lecture series on mathematical fluid dynamics in Waseda
  • Int'l Joint Research / Invited
• [Presentation] Stochastic PDE approach to random interfaces (2021)
  • Author(s): Funaki Tadahisa
  • Organizer: 信州大学理学部, zoom
  • Int'l Joint Research / Invited
• [Presentation] On the Poincaré and Related Inequalities (2020)
  • Author(s): T. Ozawa
  • Organizer: International Workshop on Multiphase Flows: Analysis, Modelling and Numerics
  • Int'l Joint Research / Invited
• [Presentation] Motion by mean curvature and coupled KPZ from particle systems. (2020)
  • Author(s): Funaki Tadahisa
  • Organizer: Asia-Pacific Analysis and PDE Seminar, University of Sydney, zoom
  • Int'l Joint Research / Invited
• [Presentation] Derivation of coupled KPZ equation from multi-species zero-range processes. (2020)
  • Author(s): Funaki Tadahisa
  • Organizer: Webinar on Stochastic Analysis, 中国科学院・北京理工大学, zoom
  • Int'l Joint Research / Invited
• [Presentation] KPZ limit for interacting particle systems. (2020)
  • Author(s): Funaki Tadahisa
  • Organizer: 北京, 清華大学丘成桐数学科学中心, Mini-Course, zoom
  • Int'l Joint Research / Invited
• [Presentation] Quasilinear stochastic PDE and its asymptotics in noise (2020)
  • Author(s): Funaki Tadahisa
  • Organizer: International Workshop on Multiphase Flows: Analysis, Modelling and Numerics, Oxford-Waseda in Mathetatics, 早稲田大学, zoom,
  • Int'l Joint Research / Invited
• [Presentation] Dirac structures in nonequilibrium thermodynamics, Keynote Lecture in Joint Structures and Common Foundation of Statistical Physics (2020)
  • Author(s): Hiroaki Yoshimura
  • Organizer: Information Geometry and Inference for Learning
  • Int'l Joint Research / Invited
• [Presentation] Dirac structures and variational formulations in nonequilibrium thermodynamics, Keynote Lecture in International Workshop on Multiphase Flows (2020)
  • Author(s): Hiroaki Yoshimura,
  • Organizer: Analysis, Modelling and Numerics Oxford-Waseda in Mathematics, Waseda University
  • Int'l Joint Research / Invited
• [Presentation] Lagrangian Coherent Structures in Rayleigh-Benard Convection with Perturbations (2020)
  • Author(s): Masahito Watanabe and Hiroaki Yoshimura
  • Organizer: International Workshop on Multiphase Flows: Analysis, Modelling and Numerics, Waseda University, Tokyo, Japan
  • Int'l Joint Research / Invited
• [Presentation] Multiphase Flow Analysis of Unsteady Behavior of Cloud Cavitation by the Smoothed Particle Hydrodynamics Method, (2020)
  • Author(s): Takahiro Ushioku and Hiroaki Yoshimura
  • Organizer: International Workshop on Multiphase Flows: Analysis, Modelling and Numerics, Waseda University, Tokyo, Japan,
  • Int'l Joint Research / Invited
• [Book] Mathematical Analysis of the Navier-Stokes Equations (2020)
  • Author(s): Govanni P. Galdi and Yoshihiro Shibata
  • Total Pages: 471
  • Publisher: Springer
  • ISBN: 978-3030362256
• [Book] Proceedings of 45th Sapporo Symposium on Partial Differential Equations (2020)
  • Author(s): S. Ei, Y. Giga, N. Hamamuki, S. Jimbo, H. Kubo, H. Kuroda, H. Kuroda, T. Ozawa, T. Sakajo, and K. Tsutaya (Eds.),
  • Total Pages: 179
  • Publisher: Hokkaido University technical report series in Mathematics
• [Book] Advances in Harmonic Analysis and Partial Differential Equations (2020)
  • Author(s): V. Georgiev, T. Ozawa, M. Ruzhansky, and J. Wirth (Eds.),
  • Total Pages: 317
  • Publisher: Trends in Mathematics, Birkhäuser
  • ISBN: 978-3-030-58214-2
• [Book] The Role of Metrics in the Theory of Partial Differential Equations (2020)
  • Author(s): Y. Giga, N. Hamamuki, H. Kubo, H. Kuroda, and T. Ozawa (Eds.),
  • Total Pages: 543
  • Publisher: Advanced Studies in Pure Mathematics, 85, 2020, Mathematical Society of Japan
• [Remarks] 流体数学研究所
  • URL: https://www.waseda.jp/inst/cro/other/2020/03/29/4693/
• [Remarks] Waseda Math-Phys
  • URL: https://www.youtube.com/channel/UCDD5lQZ4NnI15sWaDxuXylA
• [Funded Workshop] A lecture series on mathematical fluid dynamics in Waseda (2021)
• [Funded Workshop] International Workshop on Multiphase Flows Analysis, Modelling and Numerics 「Oxford-Waseda in Mathematic (2020)