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2019 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
KeywordsR-有界性 / 最大正則性原理 / 自由境界問題 / 1相問題 / 2相問題 / 多層流体 / 確率偏微分方程式 / 非平衡熱力学
Outline of Annual Research Achievements

Stokes作用素の自由境界条件下での一般化レゾルベント問題のR-有界作用素に関する一般論を完成。これを用いて時間依存問題に対する最大正則性原理を示した。Navier-Stokes方程式の1相問題の一般領域での時間局所可解性、有界領域、外部領域での時間大域解の一意存在と解の漸近挙動を示した。Stokes 作用素の2相問題に対する一般化レゾルベント問題のR-有界作用素に関する一般論を完成。これを用いて時間依存問題に対する最大正則性原理を示した。Navier-Stokes方程式の2相問題の一般領域での時間局所可解性、有界領域、外部領域での時間大域解の一意存在と解の漸近挙動を示した。複雑流体(液晶)の代表的な方程式系であるQ-tensorモデルの コーシー問題の時間大域解の一意存在を示した。流体中での化学反応に関する Stefan-Maxwell-Navier-Stokes理論により導かれる多層流体の時間局所解の存在を示した。混合系方程式の代表であるthermoelastic plate方程式に関すし最大正則性原理と半群の生成を自由境界条件の時に示した。R-有界作用素の枠組みで非斉次境界条件下での放物型方程式系の最大正則性原理を初期値境界値問題周期解について示す方法を確立した。流体の解析に現れる函数不等式を、等式の枠組で再定式化し、最適化函数の満たすべき方程式を特徴づけた。Glauber-零レンジ過程から平均曲率運動の導出を行った。非線形な拡散項を経由するため偏微分方程式の観点からも新しい結果である。非平衡熱力学に関連して,外界との間で物質及び熱の移動を伴う開放系の変分的定式化を研究し,有限次元の場合の変分構造を時間依存系の非ホロノミック系に対するラグランジュ・ダランベールの原理の観点から明らかにした。

Current Status of Research Progress
Current Status of Research Progress

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

Reason

当初の計画ではNavier-Stokes方程式の自由境界問題の時間局所解の一意存在、時間大域解の一意存在の存在を有界領域や物理的に有益な非有界領域(外部領域、Layerなど)で示すことが目的であったが、その基盤となる線形化問題の最大正則性原理を示すための一般理論の構築をR有界作用素と言う概念を導入し構成した。これは従来知られている解析半群の方法による流体数学における藤田・加藤理論を内包し準線形放物型方程式の非斉次境界条件での初期値・境界値問題の一意可解性導出のための一般理論を与える。この研究を始めた当初には予想することもなかった大理論の構築である。

Strategy for Future Research Activity

・R-有界作用素の枠組みで非斉次境界条件下での放物型方程式系の最大正則性原理を周期解について示す方法を確立する。
・電磁場閉じ込めを他の電磁場をかけることにより生じるMHD方程式の2相問題の線形化理論の確立と非線形問題の時間局所解の存在を示す。
・キャビテーション現象など気液混合の複雑な流体運動の数学的記述を可能とする方程式系として期待されているNavier-Stokes-Korteweg方程式系の時間大域解の一意存在を示す。
本年度はマクスウェル・ナビエ・ストークス系の解析を行い、2次元有界領域における時間大域解を構成する。また、流体の解析に現れる函数不等式を、等式の枠組で再定式化し、最適化函数の満たすべき方程式を特徴づける。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す。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進める。

  • Research Products

    (64 results)

All 2020 2019 Other

All Journal Article (34 results) (of which Int'l Joint Research: 31 results,  Peer Reviewed: 29 results,  Open Access: 31 results) Presentation (26 results) (of which Int'l Joint Research: 19 results,  Invited: 24 results) Book (1 results) Remarks (2 results) Funded Workshop (1 results)

  • [Journal Article] Global well-posedness for Incompressible-Incompressible two phase problem2020

    • Author(s)
      Yoshihiro Shibata Hirokazu Saito
    • Journal Title

      Fluids Under Pressure

      Volume: Chapter 3 Pages: 157-347

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the maximal Lp-Lq regulariy of solution to a general linear parabolic system2020

    • Author(s)
      Tomasz Piasecki, Yoshihiro Shibata and Ewelina Zatorska
    • Journal Title

      ournal 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] R boundedness, maximal regularity and free boundary problems for the navier stokes equations2020

    • Author(s)
      Y. Shibata
    • Journal Title

      Mathematcial Analysisi of the Navier-Stokes Equations, G.P.Galdi , Y. Shibata (Eds) C.I.M.E. Foundation Subseries @2020, Springer

      Volume: 2254 Pages: 193-462

    • DOI

      10.1007/978-3-030-36226-3_3

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Cauchy problem and vanishing dispersion limit for Schrödinger-improved Boussinesq equations2020

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

      J. Math. Anal. Appl., 485,

      Volume: Issue 2 Pages: -

    • DOI

      10.1016/j.jmaa.2020.123857

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Hardy type inequalities with spherical derivatives,2020

    • Author(s)
      N. Bez, S. Machihara, T. Ozawa
    • Journal Title

      SN Partial Differ. Equ. Appl., 1, Issue 1

      Volume: Article 5 Pages: -

    • DOI

      10.1007/s42985-019-0001-1

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Cauchy problem and vanishing dispersion limit for Schrödinger-improved Boussinesq equations2020

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

      Journal of Differential Equations, (in press).

      Volume: 268 Pages: -

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases,2020

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

      J. Math. Pures Appl., (in press).

      Volume: Volume 136 Pages: 239-256

    • DOI

      10.1016/j.matpur.2019.10.003

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On some nonlinear problem for the thermoplate equations2019

    • Author(s)
      Suma'Inna, Hirokazu Saito and Yoshihiro Shibata
    • Journal Title

      Evolution Equations and Control Theory

      Volume: 8 Pages: 755-784

    • DOI

      10.3934/eect.2019037

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the isothermal compressible multi-component mixture flow: The local existence and maximal Lp-Lq regularity of solutions2019

    • Author(s)
      Piasecki, T., Shibata, Y., Zatorska, E.
    • Journal Title

      Nonlinear Analysis, Theory, Methods and Applications

      Volume: 189 Pages: -

    • DOI

      10.1016/j.na.2019.111571

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Generation of semigroups for the thermoelastic plate equation with free boundary conditions2019

    • Author(s)
      Denk, R., Shibata, Y.
    • Journal Title

      Evolution Equations and Control Theory

      Volume: 8 Pages: 301-313

    • DOI

      10.3934/eect.2019016

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Global well-posedness and decay for a Q tensor model of incompressible nematic liquid crystals in RN2019

    • Author(s)
      Schonbek, M., Shibata, Y.
    • Journal Title

      Journal of Differential Equations

      Volume: 266 Pages: 3034-3065

    • DOI

      10.1016/j.jde.2018.08.050

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] R-solver and periodic solutons of the Navier-Stokes equations2019

    • Author(s)
      Y. Shibata
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: - Pages: -

  • [Journal Article] Fourier 変換と流体数学2019

    • Author(s)
      柴田良弘
    • Journal Title

      理工総研報告特集号16号2020年6月 ASTE Special Issue Vol. 16

      Volume: - Pages: --

  • [Journal Article] On the maximal Lp-Lq regularity theorem for the linearized electro-magnetic Magnetic field equations with interface conditions2019

    • Author(s)
      Elena Frolova and Yoshihiro Shibata
    • Journal Title

      to appear in Zapiski POMI in Russian and Journal of Mathematical Sciences

      Volume: - Pages: -

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On Strong Dynamics of Compressible Two-Component Mixture Flow2019

    • Author(s)
      Tomasz Piasecki, Yoshihiro Shibata, and Ewelina Zatorska
    • Journal Title

      SIAM Journal on Mathematical Analysis 2019

      Volume: Vol. 51, No. 4, Pages: 2793-2849

    • DOI

      10.1137/17M1151134

    • Peer Reviewed / Open Access / 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)
      Forcella, L., Fujiwara, K., Georgiev, V., Ozawa, T.
    • Journal Title

      Discrete and Continuous Dynamical Systems- Series A

      Volume: 39 Pages: 2661-2678

    • DOI

      10.3934/dcds.2019111

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] L p -Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups2019

    • Author(s)
      Ozawa, T., Ruzhansky, M., Suragan, D.
    • Journal Title

      Quarterly Journal of Mathematics

      Volume: Vol. 70, No. 1, Pages: 305-318

    • DOI

      10.1093/qmath/hay040

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case2019

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

      Trends in Mathematics

      Volume: - Pages: 179-202

    • DOI

      10.1007/978-3-030-10937-0_6

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Motion by mean curvature from Glauber-Kawasaki dynamics,2019

    • Author(s)
      Funaki, T., Tsunoda, K.
    • Journal Title

      Journal of Statistical Physics

      Volume: Vol.177 No.2 Pages: 183-208

    • DOI

      10.1007/s10955-019-02364-7

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A note on bilinear estimates in the Sobolev spaces,2019

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

      International Journal of Mathematical Analysis, 13

      Volume: no.12, Pages: 551-554

    • DOI

      10.12988/ijma.2019.91064

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A blow-up criterion for the modified Navier-Stokes-Fourier equations2019

    • Author(s)
      Jishan Fan, Tohru Ozawa
    • Journal Title

      Journal of Mathematical Fluid Mechanics, (in press). 22

      Volume: 22 Pages: -

    • DOI

      10.1007/s00021-019-0477-7

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Invariant Measures in Coupled KPZ Equations2019

    • Author(s)
      Funaki, T.
    • Journal Title

      Springer Proceedings in Mathematics and Statistics

      Volume: 282 Pages: 560-568

    • DOI

      10.1007/978-3-030-15096-9_20

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

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

      Annals of Probability

      Volume: 47 Pages: 560-612

    • DOI

      10.1214/18-AOP1268

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Fast-reaction limit for Glauber-Kawasaki dynamics with two components2019

    • Author(s)
      A.de Masi, A., Funaki, T., Presutti, E., Vares, M.E.
    • Journal Title

      Alea (Rio de Janeiro)

      Volume: Vo.16, No.2 Pages: 957-976

    • DOI

      10.30757/ALEA.v16-34

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A free energy Lagrangian variational formulation of the Navier-Stokes-Fourier system2019

    • Author(s)
      Gay-Balmaz, F., Yoshimura, H.
    • Journal Title

      International Journal of Geometric Methods in Modern Physics

      Volume: 16 Pages: -

    • DOI

      10.1142/S0219887819400061

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] From Variational to Bracket Formulations in Nonequilibrium Thermodynamics of Simple Systems2019

    • Author(s)
      Gay-Balmaz, F., Yoshimura, H.
    • Journal Title

      Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

      Volume: - Pages: 209-217

    • DOI

      10.1007/978-3-030-26980-7_22

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Dirac Structures in Open Thermodynamics2019

    • Author(s)
      Yoshimura, H., Gay-Balmaz, F.
    • Journal Title

      Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

      Volume: - Pages: 199-208

    • DOI

      10.1007/978-3-030-26980-7_21

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] From Lagrangian mechanics to nonequilibrium thermodynamics: A variational perspective2019

    • Author(s)
      Gay-Balmaz, F., Yoshimura, H.
    • Journal Title

      Entropy

      Volume: Vo.21 No.1 Pages: -

    • DOI

      10.3390/e21010008

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Design of low energy escape trajectory and delta-V reduction2019

    • Author(s)
      Tanaka, Y., Yoshimura, H., Kawakatsu, Y.
    • Journal Title

      Proceedings of the International Astronautical Congress, IAC

      Volume: - Pages: -

    • Open Access / Int'l Joint Research
  • [Journal Article] Low energy escape using tube dynamics associated with quasi-halo orbit2019

    • Author(s)
      Takemura, K., Baresi, N., Kawakatsu, Y., Yoshimura, H.
    • Journal Title

      Advances in the Astronautical Sciences

      Volume: 168 Pages: 701-711

    • Open Access / Int'l Joint Research
  • [Journal Article] Dirac structures in nonequilibrium thermodynamics2019

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

      Journal of Mathematical Physics

      Volume: 59 Pages: -

    • DOI

      10.1063/1.5017223

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Dirac structures and Lagrangian systems on tangent bundles2019

    • Author(s)
      Hiroaki Yoshimura
    • Journal Title

      Symmetry and Singularity of Geometric Structures and Differential Equations, RIMS Kokyuroku

      Volume: No.2137, Pages: -

  • [Journal Article] Proc. 4th International conference, GSI(Geometric Science and Information) 2019 at Toulouse in France, August 27-28. Lecture Notes in Computer Science2019

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

      Springer Nature Switzerland AG.

      Volume: - Pages: 209-217

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Dirac structures and port-Lagrangian systems in thermodynamics, Proc. 3rd IFAC Workshop on Thermodynamic Foundations for a Mathematical Systems Theory2019

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

      Louvain-la-Neuve

      Volume: - Pages: -

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] ポワンカレの不等式・温故知新2020

    • Author(s)
      T. Ozawa
    • Organizer
      微分方程式セミナー
    • Invited
  • [Presentation] Existence and Uniqueness of Classical Paths under Quadratic Potentials2020

    • Author(s)
      T. Ozawa
    • Organizer
      The 37th Kyushu Symposium on Partial Differential Equations
    • Int'l Joint Research / Invited
  • [Presentation] On the R-solver and periodic solutions of the Navier-Stokes equations2019

    • Author(s)
      Y. Shibata
    • Organizer
      Inhomogeneous Flows: Asymptotic Models and Interfaces Evolution
    • Int'l Joint Research / Invited
  • [Presentation] R-solver and periodic solutions of the Navier-Stokes equations2019

    • Author(s)
      Y. Shibata
    • Organizer
      非圧縮性粘性流体の数理解析, RIMS共同研究(公開型)
    • Int'l Joint Research / Invited
  • [Presentation] Self-similar solutions to the derivative nonlinear Schrödinger equation2019

    • Author(s)
      T. Ozawa
    • Organizer
      Conference on "Nonlinear Dispersive Waves, Solitons and related topics"
    • Int'l Joint Research / Invited
  • [Presentation] Self-similar solutions to the derivative nonlinear Schrödinger equation2019

    • Author(s)
      T. Ozawa
    • Organizer
      12th ISAAC Congress
  • [Presentation] 微分型相互作用をもつ非線型シュレディンガー方程式の自己相似解2019

    • Author(s)
      T. Ozawa
    • Organizer
      第9回岐阜数理科学研究会
    • Invited
  • [Presentation] Self-similar solutions to the derivative nonlinear Schrödinger equation2019

    • Author(s)
      T. Ozawa
    • Organizer
      International Conference “Actual Problems of Analysis, Differential Equations and Algebra”(National academy of sciences of the Republic of Kazakhstan)
    • Int'l Joint Research / Invited
  • [Presentation] Minimization problem on the action2019

    • Author(s)
      T. Ozawa
    • Organizer
      PDE seminar
    • Int'l Joint Research
  • [Presentation] Sharp interface limit for stochastic Allen-Cahn equations2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      2019 Spring Probability Workshop
    • Int'l Joint Research / Invited
  • [Presentation] Three different time stages in stochastic mass-conserving Allen-Cahn equation2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      Recent Trends in Stochastic Analysis and SPDEs
    • Int'l Joint Research / Invited
  • [Presentation] Large deviation for lozenge tiling dynamics2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      Stochastic Analysis, Random Fields and Integrable Probability
    • Int'l Joint Research / Invited
  • [Presentation] Scaling limits for random fields in two media2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      Gradient Flows and Related Topics: Analysis and Applications
    • Int'l Joint Research / Invited
  • [Presentation] Mean curvature interface limit from Glauber-Zero range interacting particles2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      Japanese-German Open Conference on Stochastic Analysis 2019
    • Int'l Joint Research / Invited
  • [Presentation] Stochastic mass-conserving Allen-Cahn equation, Seminaire-Groupe de Travail2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      Probabilites - Statistiques - Controle
    • Int'l Joint Research / Invited
  • [Presentation] Motion by mean curvature and coupled KPZ from particle systems2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      Mathematical Analysis for Biology and Ecology
    • Int'l Joint Research / Invited
  • [Presentation] Derivation of coupled KPZ equation from multi-species zero-range processes2019

    • Author(s)
      Tadahisa Funaki
    • Organizer
      The 18th Symposium Stochastic Analysis on Large Scale Interacting Systems
    • Int'l Joint Research / Invited
  • [Presentation] Dirac geometry, variational principles an dynamical systems in nonequilibrium thermodynamics2019

    • Author(s)
      Hiroaki Yoshimura
    • Organizer
      Workshop on Poisson Geometry and Related Topics
    • Int'l Joint Research / Invited
  • [Presentation] Experimental observations of single bubble dynamics and induced shock waves2019

    • Author(s)
      Takahiro Ushioku, Masaki Kunishima, Takashi Imamura and Hiroaki Yoshimura
    • Organizer
      4th International Conference on Recent Advances in Nonlinear Mechanics (RANM 2019)
    • Int'l Joint Research / Invited
  • [Presentation] Chaotic mixing in two-dimensional Rayleigh-Benard convection with periodic perturbations2019

    • Author(s)
      Masahito Watanabe and Hirokai Yoshimura
    • Organizer
      4th International Conference on Recent Advances in Nonlinear Mechanics (RANM 2019
    • Int'l Joint Research / Invited
  • [Presentation] Continuous and discrete variational formulations in nonequilibrium thermodynamics2019

    • Author(s)
      Hiroaki Yoshimura and Francois Gay-Balmaz
    • Organizer
      AMS Spring Central and Western Joint Sectional Meeting, Special Session on Geometric Approaches to Mechanics and Control IV
    • Int'l Joint Research / Invited
  • [Presentation] Dirac structures and dynamical systems in nonequilibrium thermodynamics2019

    • Author(s)
      Francois Gay-Balmaz and Hiroaki Yoshimura
    • Organizer
      AMS Spring Central and Western Joint Sectional Meeting, Special Session on Geometric Approaches to Mechanics and Control IV
    • Int'l Joint Research / Invited
  • [Presentation] 摂動を受けるレイリー・ベナール対流に現れるカオス的混合と分岐現象2019

    • Author(s)
      渡辺昌仁, 吉村浩明
    • Organizer
      日本機械学会 Dynamics and Design Conference 2019
    • Invited
  • [Presentation] 摂動を受けるレイリー・ベナール対流に現れるカオス的混合と分岐現象2019

    • Author(s)
      渡辺昌仁, 吉村浩明
    • Organizer
      日本応用数理学会講演
    • Invited
  • [Presentation] 非平衡熱力学の変分的定式化と幾何学的構造2019

    • Author(s)
      吉村浩明
    • Organizer
      研究会「量子と古典の物理と幾何」東工大学先端化学技術センター
    • Invited
  • [Presentation] 「非平衡熱力学の理解へ向けて ― 変分原理と幾何学的アプローチ ―」2019

    • Author(s)
      吉村浩明
    • Organizer
      計測自動制御学会 制御部門,真なるダイナミクスの追求による次世代システム制御理論調査研究会
    • Invited
  • [Book] 『確率偏微分方程式』数学叢書2019

    • Author(s)
      舟木直久, 乙部厳己, 謝賓,
    • Total Pages
      350
    • Publisher
      岩波書店
    • ISBN
      9784000298254
  • [Remarks] スーパーグローバル大学創成支援 早稲田大学数物系科学拠点

    • URL

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

  • [Remarks] Institute of Mathematical Fluid Dynamics 流体数学研究所

    • URL

      https://www.waseda.jp/inst/cro/other/2020/03/29/4693/

  • [Funded Workshop] International Workshop on Multi-Phase Flows: Analysis, Modelling and Numerics2019

URL: 

Published: 2021-12-27  

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