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2018 Fiscal Year Annual Research Report

Construction of Mathematical Theory of 2 phase fluid flows

Research Project

Project/Area Number 17H01097
Research InstitutionWaseda 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 / 数値シュミレーション
Outline of Annual Research Achievements

1. Navier-Stokes-Koretweg 方程式を基盤とした相転移を伴う2相問題について、一般領域でその線形化問題に対するLp―Lq最大正則性原理を対応するレゾルベント問題のR有界な解作用素を構成することで示した。また有界領域で線形化問題の解の指数安定性を示した。これらの結果を基盤として非線形問題の時間局所解、時間大域解の一意存在を示した。
またNavier-Stokes方程式の自由境界問題を外部領域で考え表面張力が無い場合の時間大域解の一意存在を示した。またNavier-Stokes方程式の2相問題をjump面が有界領域で表面張力が考慮された問題を全空間で考え時間大域解の一意存在を示した。
3.Navier-Stokes方程式以外の流体系の方程式として, Nemaric Liquid Crystal のQ-tensor model を全空間で考えその時間大域解の一意存在を示した。
4.ミクロからの流体現象の解析として、平衡熱力学系の変分的定式化を開放系に適用できるように拡張を行った.閉鎖的な非平衡熱力学系の変分的定式化は、非線形かつ非ホロノミックな拘束を受けるラグランジュ・ダランベールの原理から導かれることを明らかにしているが、この変分的理論を開放系拡張するために、時間依存の非線形かつ非ホロノミックな拘束を受けるラグランジュ.ダランベールの原理を提案することに成功した。また相互作用粒子系から平均曲率運動,Stefan 自由境界問題,発散項を含む特異な確率偏微分方程式である多成分KPZ方程式などの導出を行った。
5. 数値解析的アプローチとしては, 相変化を伴う圧縮性二相流れに対するNavier-Stokes-Korteweg方程式を、歪対称のPoisson括弧と半負定値対称の散逸括弧を用いて定式化した。

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

1. 「6.研究の概要」で述べたことは2018年度の申請時における構想をほぼ実現化したものであり、研究計画には無理がなく研究は順調に進展している。
2.磁気流体の2相問題、多成分流体の方程式系についての時間局所解、大域解の存在を示すための知見を得た。また流体方程式の自由境界問題に対する周期解の存在はこれまで全く手のついてない問題であったが、R有界作用素も用いることで高周波部分の最大正則性原理を得られる知見を得た。さらに液滴落下の問題もその定常解の存在に関しては見通しがついた。こうして、2019年度において磁気流体、多成分流体、液滴落下問題に対しての研究の進展がおおいに見込まれる。
4.自由境界面を扱う数値解析は非常に困難であるが、2018年度よりアイソジオメトリック解析による方法の開発に取り組んでいる。
4.国際的な研究協力による研究の進展という観点からは、2018年度も早稲田大学で数値流体と流体数学がコラボした国際研究集会を行った。また非圧縮性粘性流体に話題を限った国際研究集会を京都大学数理解析研究所で開催した。早稲田大学では述べ100名程度、京都大学ではのべ70名程度の参加者を得、最先端の講演と活発な討論がなされた。
5.若手研究者育成という面では、SGU数物系拠点において博士後期課程が2年目を迎え、数学・物理・機械工学・数値流体を専門とする計8名の大学院生がこの博士後期課程に現在在籍し、講義のみならず、3の国際研究集会の参加・発表、海外留学を経験し、国際的・分野横断的教育を受け多くの成果を挙げた。特にロンドンインペリアルカレッジ、ハンブルグ大学とは大学協定、ピサ大学とはコチュテルを結び国際協力のもと学生の共同研究指導名を充実させる方向に展開している。

Strategy for Future Research Activity

1. マクロの視点からの流体解析研究では、多成分流体方程式、磁気流体方程式の時間局所解、大域解の一意存在をR有界な解作用素とスペクトル解析の観点から進める。またR有界な解作用素理論を基盤として、Navier-Stokes方程式の自由境界問題、2相問題に対する周期解の存在の研究を進める。さらに液滴落下問題の定常解の存在証明を完成し、また非定常問題の解析へと進展させる。
2.ミクロの視点からの流体解析研究では、概ね当初の研究計画通りに研究が進展しているので引き続き研究を継続し、理論形成の発展を目指す。
3.数値流体は早稲田大学の学内組織として立ち上げた熱変換工学・数学研究所が3年目を迎え、工学との融合研究が展開しているのでその流れの中で、数学からモデリングを提唱するという立場で特にアイソジオメトリック法の基づく研究をすすめていく。また豊田中研との共同研究も2018年度より開始したので、流体数学研究の出口研究を共同で行っていく。

  • Research Products

    (66 results)

All 2019 2018 Other

All Int'l Joint Research (5 results) Journal Article (21 results) (of which Int'l Joint Research: 18 results,  Peer Reviewed: 21 results,  Open Access: 2 results) Presentation (35 results) (of which Int'l Joint Research: 20 results,  Invited: 35 results) Book (1 results) Remarks (2 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Department of Mathematics/University of Pittsburgh(米国)

    • Country Name
      U.S.A.
    • Counterpart Institution
      Department of Mathematics/University of Pittsburgh
  • [Int'l Joint Research] Department of Mathematics/University of Oxford(英国)

    • Country Name
      UNITED KINGDOM
    • Counterpart Institution
      Department of Mathematics/University of Oxford
  • [Int'l Joint Research] Department of Mathematics/University of Warsaw(ポーランド)

    • Country Name
      POLAND
    • Counterpart Institution
      Department of Mathematics/University of Warsaw
  • [Int'l Joint Research] Department of Mathematics/Technical University of Darmstadt(ドイツ)

    • Country Name
      GERMANY
    • Counterpart Institution
      Department of Mathematics/Technical University of Darmstadt
  • [Int'l Joint Research] Steklov Institute of Mathematics(ロシア連邦)

    • Country Name
      RUSSIA FEDERATION
    • Counterpart Institution
      Steklov Institute of Mathematics
  • [Journal Article] Global well-posedness and decay for a Q tensor model of incompressible namatic liquid crystals in RN2019

    • Author(s)
      Maria Schonbek and Yoshihiro Shibata
    • Journal Title

      Journal of Differential Equations

      Volume: 266 Pages: 3034―3065

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Sharp interface limit for stochastically perturbed mass conserving  Allen-Cahn equation2019

    • Author(s)
      T. Funaki and S. Yokoyama
    • Journal Title

      Ann. Probab

      Volume: 47ページ Pages: 560-612

    • DOI

      10.1214/18-AOP1268

    • Peer Reviewed / Open Access
  • [Journal Article] 長田博文氏の業績---長距離相互作用を持つ無限粒子系の確率解析---2019

    • Author(s)
      舟木直久
    • Journal Title

      数学

      Volume: 71 Pages: -

    • Peer Reviewed
  • [Journal Article] Note for global existence of semilinear heat equation in weighted L∞2019

    • Author(s)
      K. Fujiwara, V. Georgiev, T. Ozawa
    • Journal Title

      Pliska Stud. Math

      Volume: 30 Pages: 7-20

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential2019

    • Author(s)
      L. Forcella, K. Fujiwara, V. Georgiev, T. Ozawa
    • Journal Title

      Discrete and Continuous Dynamical Systems

      Volume: 39 Pages: 2661-2678

    • DOI

      10.3934/dcds.2019111

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the Lp-Lq decay estimate for the Stokes equations with free boundary conditions in an exterior domain2018

    • Author(s)
      Yoshihiro Shibata
    • Journal Title

      Asymptotic Analysis

      Volume: 107 Pages: 33--72

    • DOI

      10.3233/ASY-171449

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain2018

    • Author(s)
      Yoshihiro Shibata
    • Journal Title

      Communication on Pure and Applied Analysis

      Volume: 17 Pages: 1681-172

    • DOI

      0.3934/cpaa.2018081

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface2018

    • Author(s)
      Yoshihiro Shibata
    • Journal Title

      Evolution Equations and Control Theory

      Volume: 7 Pages: 117-152

    • DOI

      10.3934/eect.2018007

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the maximal Lp-Lq theory arising in the study of a free boundary problem for the Navier-Stokes equations2018

    • Author(s)
      Yoshihiro Shibata and Suma’Inna
    • Journal Title

      FMRIJ-18-eBook-220

      Volume: - Pages: -

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A ariational formula of nonequilibrium thermodynamics for discrete opern systems with mass and heat transfer2018

    • Author(s)
      Francois Gay-Balmaz and Hiroaki Yoshimura
    • Journal Title

      Entropy

      Volume: 163 Pages: -

    • DOI

      10.3390/e20030163

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Dirac structures in nonequilibrium thermodynamics2018

    • Author(s)
      Francois Gay-Balmaz and Hiroaki Yoshimura
    • Journal Title

      J. Math. Phys

      Volume: 59 Pages: 1673-1705

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Variational discretization for the nonlequilibrium thermodynamics of simple system2018

    • Author(s)
      Francois Gay-Balmaz and Hiroaki Yoshimura
    • Journal Title

      Nonlinearity

      Volume: 31 Pages: 1673-1705

    • DOI

      10.1007/978-3-319-74929-7_6

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Curvature motion perturbed by a direction-dependent colored noise2018

    • Author(s)
      C. Denis, T. Funaki and S. Yokoyama
    • Journal Title

      Stochastic Partial Differential Equations and Related Fields, Springer Proceedings in Mathematics & Statistics

      Volume: 229 Pages: 177-200

    • DOI

      10.1007/978-3-319-74929-7_6

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Convergence of a finite volume scheme for a stochastic conservation law involving a Q-Brownian motion,Discrete Contin2018

    • Author(s)
      T. Funaki, Y. Gao and D. Hilhorst
    • Journal Title

      Discrete Contin. Dyn. Syst. Ser. B,

      Volume: 23 Pages: 1459-1502

    • DOI

      org/10.3934/dcdsb.2018159

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Dynamical behavior for the solutions of the Navier-Stokes equation2018

    • Author(s)
      K. Li, T. Ozawa, B. Wang
    • Journal Title

      Pure Appl. Anal

      Volume: 17時まで Pages: 1511-1560

    • DOI

      10.3934/cpaa.2018073

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Stability of trace theorems on the sphere2018

    • Author(s)
      N. Bez, C. Jeavons, T. Ozawa, M. Sugimoto
    • Journal Title

      J. Geom. Anal

      Volume: 28 Pages: 1456-1476

    • DOI

      10.1007/s12220-017-9870-8

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation2018

    • Author(s)
      K. Fujiwara, T. Ozawa
    • Journal Title

      Evolution Equations and Control Theory

      Volume: 7 Pages: 275-280

    • DOI

      10.3934/eect.2018013

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Higher order fractional Leibniz rule2018

    • Author(s)
      K. Fujiwara, V. Georgiev, T. Ozawa
    • Journal Title

      J. Fourier Anal. Appl

      Volume: 24 Pages: 650-665

    • DOI

      10.1007/s00041-017-9541-y

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Blow-up for self-interacting fractional Ginzburg-Landau equation2018

    • Author(s)
      K. Fujiwara, V. Georgiev, T. Ozawa
    • Journal Title

      Dynamics of PDE

      Volume: 15 Pages: 175-182

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Global well-posedness of weak solutions to the time-dependent Ginzburg-Landau model for superconductivity2018

    • Author(s)
      J. Fan, T. Ozawa
    • Journal Title

      Taiwanese J. Math

      Volume: 22 Pages: 851-858

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Compressible-incompressible two-phase flows with phase transition: model problem2018

    • Author(s)
      K. Watanabe
    • Journal Title

      J. Math. Fluid Mech

      Volume: 20 Pages: 969-1011

    • Peer Reviewed
  • [Presentation] R boundedness, maximal regularity and one phase problem for the NS eq2019

    • Author(s)
      柴田良弘
    • Organizer
      RIMS共同研究(公開型)Maximal regularity and nonlinear PDE
    • Invited
  • [Presentation] Hydrodynamic limits for stochastic systems2019

    • Author(s)
      舟木直久
    • Organizer
      Gran Sasso Science Institute, Series of Lectures
    • Invited
  • [Presentation] Stochastic PDE approach to random interfaces2019

    • Author(s)
      舟木直久
    • Organizer
      Spring School on Random Interfaces Pisa-Augsburg School, Series of Lectures
    • Int'l Joint Research / Invited
  • [Presentation] erivation of coupled KPZ-Burgers equation from multi-species zero-range processes2019

    • Author(s)
      舟木直久
    • Organizer
      New Directions in Stochastic Analysis: Rough Paths, SPDEs and Related Topics
    • Int'l Joint Research / Invited
  • [Presentation] 微分型シュレディンガー方程式の自己相似解2019

    • Author(s)
      小澤徹
    • Organizer
      PDE Workshop in Miyazaki
    • Invited
  • [Presentation] Cauchy problem for the quasilinear evolution equation of transverse wave model2019

    • Author(s)
      小澤徹
    • Organizer
      PDE Workshop Waseda - GSSI, L'Aquila-Pisa
    • Invited
  • [Presentation] On some free boundary problem for the Navier-StoWorkshop on Mathematical Fluid Dynamics in2018

    • Author(s)
      柴田良弘
    • Organizer
      Bad Bollkes equations
    • Int'l Joint Research / Invited
  • [Presentation] 2 phase problem for the Navier-Stokes equations in the whole space2018

    • Author(s)
      柴田良弘
    • Organizer
      PDEs and Mathematical Hydrodynamics in Honor of Vsevolod Alekseevich
    • Int'l Joint Research / Invited
  • [Presentation] 2 phase problem for the Navier-Stokes equations in the whole space2018

    • Author(s)
      柴田良弘
    • Organizer
      Mathflows
    • Int'l Joint Research / Invited
  • [Presentation] A variational formulation, Dirac structures and dynamical systems for nonequilibrium thermodynamics2018

    • Author(s)
      吉村浩明
    • Organizer
      Workshop of Symmetry and Singularity of Geometric Structures and Differential Equations
    • Invited
  • [Presentation] Dirac structures in thermodynamics2018

    • Author(s)
      吉村浩明
    • Organizer
      IFAC 6th Workshop on Lagrangian and Hamiltonian nonlinear control-LHMNC2018
    • Int'l Joint Research / Invited
  • [Presentation] Dirac structures, interconnections, and variational formulations in nonequilibrium thermodynamics, Entropy 20182018

    • Author(s)
      吉村浩明
    • Organizer
      From Physics to Information Sciences and Geometry, Special session: Fourier 250th Birthday: Geometric Theory of Thermodynamics
    • Int'l Joint Research / Invited
  • [Presentation] Mixing and Lagrangian coherent structures in Two-dimensional Rayleigh-Benard convection with periodic perturbations2018

    • Author(s)
      吉村浩明(Masahito Watanabe, Tomohiro Miyamoto)
    • Organizer
      SIAM Conference on Nonlinear Waves and Coherent Structures
    • Invited
  • [Presentation] Design of low energy Earth-Moon transfers in the 4-body system2018

    • Author(s)
      吉村浩明
    • Organizer
      AIMS 2018 Special session 25: Celestial mechanics and N-body problem
    • Invited
  • [Presentation] Design of low energy Earth-Moon transfers in the 4-body system2018

    • Author(s)
      吉村浩明
    • Organizer
      AIMS 2018 Special session 147: Structure preserving numerical methods
    • Int'l Joint Research / Invited
  • [Presentation] A variational formulation, Dirac structures and dynamical systems for nonequilibrium thermodynamics2018

    • Author(s)
      吉村浩明
    • Organizer
      International Conference on Geometric Mechanics and Control
    • Int'l Joint Research / Invited
  • [Presentation] The discrete Lagrange-d'Alembert principle for interconnected systems with constraints2018

    • Author(s)
      吉村浩明(Linyu Peng)
    • Organizer
      ICDVC-2018
    • Int'l Joint Research / Invited
  • [Presentation] Geometry, Variational Formulations and Interconnection for Physical System Dynamics, Toward understanding multi-physical systems (Keynote lecture)2018

    • Author(s)
      吉村浩明
    • Organizer
      ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
    • Int'l Joint Research / Invited
  • [Presentation] Derivation of coupled KPZ equation from multi-color zero-range processes2018

    • Author(s)
      舟木直久
    • Organizer
      Motion by mean curvature from Glauber-Kawasaki dynamics
    • Int'l Joint Research / Invited
  • [Presentation] Motion by mean curvature from Glauber-Kawasaki dynamics2018

    • Author(s)
      舟木直久
    • Organizer
      Workshop on Interplay of Random Media and Stochastic Interface Models
    • Invited
  • [Presentation] Coupled KPZ (Kardar-Parisi-Zhang) equation2018

    • Author(s)
      舟木直久
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations 
    • Int'l Joint Research / Invited
  • [Presentation] Motion by mean curvature from Glauber-Kawasaki dynamics2018

    • Author(s)
      舟木直久
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations 
    • Invited
  • [Presentation] Derivation of coupled KPZ equation from multi-color zero-range process2018

    • Author(s)
      舟木直久
    • Organizer
      Workshop on Stochastic partial differential equations and related topics
    • Invited
  • [Presentation] Hydrodynamic limit for exclusion processes2018

    • Author(s)
      舟木直久
    • Organizer
      KAIST Summer School in Probability 2018, Series of Lectures
    • Int'l Joint Research / Invited
  • [Presentation] Coupled KPZ equation2018

    • Author(s)
      舟木直久
    • Organizer
      2018 Probability Workshop in Gyeongju
    • Invited
  • [Presentation] Derivation of motion by mean curvature and Stefan problem from particle2018

    • Author(s)
      舟木直久
    • Organizer
      ReaDiNet 2018 Recent Progresses in Mathematical Theories for 
    • Invited
  • [Presentation] Derivations of motion by mean curvature and Stefan problem from Glauber -Kawasaki dynamics2018

    • Author(s)
      舟木直久
    • Organizer
      17th workshop on Stochastic Analysis on Large Scale Interacting Systems
    • Invited
  • [Presentation] On improved Hardy inequalities2018

    • Author(s)
      小澤徹
    • Organizer
      Workshop in Hangzhou 2018
    • Int'l Joint Research / Invited
  • [Presentation] Improved Hardy inequalities2018

    • Author(s)
      小澤徹
    • Organizer
      PDE Workshop
    • Int'l Joint Research / Invited
  • [Presentation] Lifespan estimates of solutions to NLS without gauge invariance2018

    • Author(s)
      小澤徹
    • Organizer
      PDE Seminar
    • Int'l Joint Research / Invited
  • [Presentation] Improved Hardy inequalities Celebrating Approximate 60s2018

    • Author(s)
      小澤徹
    • Organizer
      An International Conference on Nonlinear PDEs and Its Applications at NYU Shanghai
    • Int'l Joint Research / Invited
  • [Presentation] Lifespan of blowup solutions of DNLS type equation on the torus2018

    • Author(s)
      小澤徹
    • Organizer
      PDE Seminar
    • Int'l Joint Research / Invited
  • [Presentation] さまざまなハーディー型不等式2018

    • Author(s)
      小澤徹
    • Organizer
      作用素論セミナー
    • Invited
  • [Presentation] Improved Hardy inequalities2018

    • Author(s)
      小澤徹
    • Organizer
      Nonlinear Dispersive Equations at Florianopolis
    • Int'l Joint Research / Invited
  • [Presentation] Improved Hardy inequalities2018

    • Author(s)
      小澤徹
    • Organizer
      Mathematical Fluid Mechanics and Related Topics - in honor of Professor Hideo
    • Invited
  • [Book] 確率偏微分方程式2019

    • Author(s)
      舟木 直久、乙部 厳己、謝 賓
    • Total Pages
      376
    • Publisher
      岩波書店
    • ISBN
      9784000298254
  • [Remarks] 柴田研究室

    • URL

      http://www.fluid.sci.waseda.ac.jp/shibata/top.html

  • [Remarks] スーパーグローバル大学創生支援、早稲田大学数物系科学拠点

    • URL

      http://www.sgu-mathphys.sci.waseda.ac.jp/

  • [Funded Workshop] International Workshop on Multi-Phase Flows: Analysis, Modelling and Numerics2018

  • [Funded Workshop] International Workshop on “Fundamental Problems in Mathematical and Theoretical Physics”2018

URL: 

Published: 2019-12-27  

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