Iguchi, Tatsuo

写真a

Affiliation

Faculty of Science and Technology, Department of Mathematics (Yagami)

Position

Professor

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Career 【 Display / hide 】

  • 1997.01
    -
    2002.03

    九州大学大学院数理学研究科文部教官助手

  • 2002.04
    -
    2006.03

    東京工業大学大学院理工学研究科文部科学教官助教授

  • 2006.04
    -
    2011.03

    慶應義塾大学理工学部助教授

  • 2011.04
    -
    Present

    慶應義塾大学理工学部教授

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Academic Background 【 Display / hide 】

  • 1993.03

    Waseda University, Faculty of Science and Engineering, Department of Mathematics

    University, Graduated

  • 1995.04

    Waseda University, Graduate School, Division of Science and Engineering, 数理科学専攻

    Graduate School, Completed, Master's course

  • 1996.12

    Waseda University, Graduate School, Division of Science and Engineering, 数理科学専攻

    Graduate School, Withdrawal before completion, Doctoral course

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Academic Degrees 【 Display / hide 】

  • On the Well-Posedness of Initial Value Problems for Ideal Fluid with Free Boundary, Waseda University, Dissertation, 1998.03

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Research Areas 【 Display / hide 】

  • Natural Science / Basic analysis (Basic Analysis)

  • Natural Science / Mathematical analysis (Global Analysis)

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Papers 【 Display / hide 】

  • Well-posedness of the initial boundary value problem for degenerate hyperbolic systems with a localized term and its application to the linearized system for the motion of an inextensible hanging string

    Iguchi,T., Takayama M.

    Osaka Journal of Mathematics  2025

    Research paper (scientific journal), Accepted,  ISSN  0030-6126

    　View Summary

    Motivated by an analysis on the well-posedness of the initial boundary value problem for the motion of an inextensible hanging string, we first consider an initial boundary value problem for one-dimensional degenerate hyperbolic systems with a localized term and show its well-posedness in weighted Sobolev spaces. We then consider the linearized system for the motion of an inextensible hanging string. Well-posedness of its initial boundary value problem is demonstrated as an application of the result obtained in the first part.

  • The 2D nonlinear shallow water equations with a partially immersed obstacle

    Iguchi, T., Lannes D.

    Journal of the European Mathematical Society (The Publishing House of the European Mathematical Society)   2025

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  1435-9855

    　View Summary

    This article is devoted to the proof of the well-posedness of a model describing waves propagating in shallow water in horizontal dimension d=2 and in the presence of a fixed partially immersed object. We first show that this wave-interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain, with boundary conditions that are fully nonlinear and nonlocal in space and time. This hyperbolic initial boundary value problem is characteristic, does not satisfy the constant rank assumption on the boundary matrix, and the boundary conditions do not satisfy any standard form of dissipativity. Our main result is the well-posedness of this system for irrotational data and at the quasilinear regularity threshold. In order to prove this, we introduce a new notion of weak dissipativity, that holds only after integration in time and space. This weak dissipativity allows higher order energy estimates without derivative loss; the analysis is carried out for a class of linear non-characteristic hyperbolic systems, as well as for a class of characteristic systems that satisfy an algebraic structural property that allows us to define a generalized vorticity. We then show, using a change of unknowns, that it is possible to transform the linearized wave-interaction problem into a non-characteristic system, which satisfies this structural property and for which the boundary conditions are weakly dissipative. We can therefore use our general analysis to derive linear, and then nonlinear, a priori energy estimates. Existence for the linearized problem is obtained by a regularization procedure that makes the problem non-characteristic and strictly dissipative, and by the approximation of the data by more regular data satisfying higher order compatibility conditions for the regularized problem. Due to the fully nonlinear nature of the boundary conditions, it is also necessary to implement a quasilinearization procedure. Finally, we have to lower the standard requirements on the regularity of the coefficients of the operator in the linear estimates to be able to reach the quasilinear regularity threshold in the nonlinear well-posedness result.

  • A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: Justification as a shallow water approximation

    Duchêne V., Iguchi T.

    Proceedings of the Royal Society of Edinburgh Section A: Mathematics (Cambridge University Press)   2024.03

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  0308-2105

    　View Summary

    We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler-Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order in the sense of consistency, where and are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.

  • A priori estimates for solutions to equations of motion of an inextensible hanging string

    Iguchi T., Takayama M.

    Mathematische Annalen (Springer)  390   1919 - 1971 2024.01

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  0025-5831

    　View Summary

    We consider the initial boundary value problem to equations of motion of an inextensible hanging string of finite length under the action of the gravity. We also consider the problem in the case without any external forces. In this problem, the tension of the string is also an unknown quantity. It is determined as a unique solution to a two-point boundary value problem, which is derived from the inextensibility of the string together with the equation of motion, and degenerates linearly at the free end. We derive a priori estimates for solutions to the initial boundary value problem in weighted Sobolev spaces under a natural stability condition. The necessity for the weights results from the degeneracy of the tension. Uniqueness of solutions is also proved.

  • A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part I: Structures and well-posedness

    Duchêne V., Iguchi T.

    Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire (The Publishing House of the European Mathematical Society)  41 ( 2 ) 257 - 315 2023.03

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  0294-1449

    　View Summary

    We consider a model, which we named the Kakinuma model, for interfacial gravity waves. As is well known, the full model for interfacial gravity waves has a variational structure whose Lagrangian is an extension of Luke’s Lagrangian for surface gravity waves, that is, water waves. The Kakinuma model is a system of Euler–Lagrange equations for approximate Lagrangians, which are obtained by approximating the velocity potentials in the Lagrangian for the full model. In this paper we first analyze the linear dispersion relation for the Kakinuma model and show that the dispersion curves highly fit that of the full model in the shallow water regime. We then analyze the linearized equations around constant states and derive a stability condition, which is satisfied for small initial data when the denser water is below the lighter water. We show that the initial value problem is in fact well posed locally in time in Sobolev spaces under the stability condition, the noncavitation assumption, and intrinsic compatibility conditions, in spite of the fact that the initial value problem for the full model does not have any stability domain so that its initial value problem is ill posed in Sobolev spaces. Moreover, it is shown that the Kakinuma model enjoys a Hamiltonian structure and has conservative quantities: mass, total energy, and in the case of a flat bottom, momentum.

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Papers, etc., Registered in KOARA 【 Display / hide 】

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Reviews, Commentaries, etc. 【 Display / hide 】

  • The water wave equations

    Iguchi T.

    Sugaku Expositions (American Mathematical Society)  35 ( 1 ) 53 - 81 2022.04

    Article, review, commentary, editorial, etc. (scientific journal), Single Work,  ISSN  0898-9583

  • 水の波の方程式

    井口 達雄

    雑誌『数学』 (日本数学会)  70 ( 1 ) 1 - 25 2018.01

    Article, review, commentary, editorial, etc. (scientific journal), Single Work,  ISSN  0039-470X

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Presentations 【 Display / hide 】

  • 吊り下げられた紐の運動に対する初期境界値問題の適切性

    高山 正宏，井口 達雄

    日本数学会年会 (早稲田大学) , 

    2025.03
    , 

    Oral presentation (general), 日本数学会

  • 局所化項をもつ退化双曲系に対する初期境界値問題の適切性

    高山 正宏，井口 達雄

    日本数学会秋季総合分科会 (大阪大学) , 

    2024.09
    , 

    Oral presentation (general), 日本数学会

  • 吊り下げられた紐の運動に対する線形化問題の適切性

    高山 正宏，井口 達雄

    日本数学会秋季総合分科会 (大阪大学) , 

    2024.09
    , 

    Oral presentation (general), 日本数学会

  • 吊り下げられた紐の運動に対する解のアプリオリ評価

    高山 正宏，井口 達雄

    日本数学会年会 (大阪公立大学) , 

    2024.03
    , 

    Oral presentation (general), 日本数学会

  • A mathematical analysis of the Kakinuma model for interfacial gravity waves

    井口達雄

    RIMS共同研究（公開型）「流体と気体の数学解析」 (京都大学数理解析研究所 (online)) , 

    2021.07
    , 

    Oral presentation (general)

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Research Projects of Competitive Funds, etc. 【 Display / hide 】

  • 水の波の数学解析の新展開

    2022.04
    -
    2027.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, 基盤研究(B), Principal investigator

  • 水の波の新しいモデルの創出とその数学解析

    2017.06
    -
    2020.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Challenging Research (Exploratory) , Principal investigator

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Awards 【 Display / hide 】

  • IOP Outstanding Reviewer Awards 2021 (Journal of Physics A: Mathematical and Theoretical)

    2022.04

    Type of Award： Honored in official journal of a scientific society, scientific journal

  • IOP Outstanding Reviewer Awards 2018 (Nonlinearity)

    2019.03

    Type of Award： Honored in official journal of a scientific society, scientific journal

  • 日本数学会 函数方程式論分科会 福原賞

    IGUCHI Tatsuo, 2010.12, 水面波方程式の数学解析の研究

    Type of Award： Award from Japanese society, conference, symposium, etc.

  • 手島工業教育資金団藤野研究賞

    IGUCHI Tatsuo, 2003.03, 水面波の方程式の解析的研究

    Type of Award： Award from publisher, newspaper, foundation, etc.

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Courses Taught 【 Display / hide 】

  • PRINCIPLES OF FUNCTIONAL EQUATIONS

    2025

  • MATHEMATICS 2B

    2025

  • MATHEMATICS 2A

    2025

  • MATHEMATICAL ANALYSIS 2

    2025

  • INDEPENDENT STUDY ON FUNDAMENTAL SCIENCE AND TECHNOLOGY

    2025

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Memberships in Academic Societies 【 Display / hide 】

  • 日本数学会, 

    1994
    -
    Present
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