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Kubo Hideo

Faculty of Science　Mathematics　MathematicsProfessor
Research Center of Mathematics for Social CreativityProfessor

Researcher basic information

■ Degree
  • 博士（理学）, 北海道大学
■ URL
researchmap URL■ Various IDs
J-Global ID■ Research Keywords and Fields
Research Keyword
  • 散乱理論
  • 非線型波動
  • Scattering theory
  • Nonlinear Wave
Research Field
  • Natural Science, Basic analysis
■ Educational Organization

Career

■ Career
Career
  • Oct. 2012 - Present
    Hokkaido University, 教授
  • Oct. 2008 - Sep. 2012
    Tohoku University, 教授
  • Apr. 2003 - Sep. 2008
    Osaka University, 助教授
  • Apr. 1997 - Mar. 2003
    Shizuoka University
  • Apr. 1996 - Mar. 1997
    Shizuoka University
Educational Background
  • Mar. 1996, Hokkaido University, 理学研究科, 数学専攻, Japan
  • 1991, Hokkaido University, School of Science, 数学, Japan
Committee Memberships
  • Mar. 2025 - Present
    日本数学会 評議員
  • Jun. 2021 - Present
    日本数学会, 理事, Society
  • 2017 - Present
    日本数学会 函数方程式論分科会, 委員, Society
  • Mar. 2021 - Feb. 2025
    日本数学会 函数方程式論分科会, 評議員, Society
  • 2009 - 2010
    日本数学会, 東北支部 代議員, Society
  • 2008 - 2009
    日本数学会, 「数学通信」編集委員, Society
  • 2008 - 2009
    日本数学会, 東北支部 連絡責任評議員, Society
Position History
  • 大学院理学院副学院長, 2021年4月1日 - 2023年3月31日
  • 大学院理学院副学院長, 2023年4月1日 - 2025年3月31日
  • 大学院理学研究院副研究院長, 2023年4月1日 - 2025年3月31日

Research activity information

■ Awards
  • 2002, 日本数学会賞建部賢弘 特別賞
    Japan
■ Papers
■ Other Activities and Achievements
■ Books and other publications
  • 多変数の微積分とベクトル解析 (新・数理/工学ライブラリ 応用数学 3)
    神保 秀一; 久保 英夫
    数理工学社, 10 Sep. 2020, 4864810680, 176, Japanese, [Joint work]
  • The role of metrics in the theory of partial differential equations, Advanced Studies in Pure Mathematics, 85
    Y. Giga; N. Hamamuki; H. Kuroda; T. Ozawa; H. Kubo
    Mathematical Society of Japan, 2020, 9784864970907, 543p, English, Scholarly book, [Joint editor]
  • RIMS Kôkyûroku Bessatsu B70 "Harmonic Analysis and Nonlinear Partial Differential Equations"
    Hideo Takaoka; Hideo Kubo
    Research Institute for Mathematical Sciences Kyoto University, Apr. 2018, 166, [Joint editor]
  • RIMS Kôkyûroku Bessatsu B65 "Harmonic Analysis and Nonlinear Partial Differential Equations"
    Hideo Kubo; Hideo Takaoka
    Research Institute for Mathematical Sciences Kyoto University, May 2017
  • RIMS Kôkyûroku Bessatsu B60 "Harmonic Analysis and Nonlinear Partial Differential Equations"
    Hideo Kubo; Mitsuru Sugimoto
    Research Institute for Mathematical Sciences Kyoto University, Dec. 2016, 212, [Joint editor]
  • RIMS Kôkyûroku Bessatsu B56 "Harmonic Analysis and Nonlinear Partial Differential Equations"
    Hideo Kubo; Mitsuru Sugimoto
    Research Institute for Mathematical Sciences Kyoto University, Apr. 2016, 215, [Joint editor]
  • RIMS Kôkyûroku Bessatsu B49 "Harmonic Analysis and Nonlinear Partial Differential Equations"
    Hideo Kubo; Mitsuru Sugimoto
    Research Institute for Mathematical Sciences Kyoto University, Apr. 2014, 137
  • Hokkaido Math. J. vol.37
    Hideo Kubo; Hiroyuki Takamura, Special Issue “Nonlinear Wave Equations”
    Hokkaido University, 2008, [Joint editor]
  • "New trends in the theory of hyperbolic equations", Oper. Theory Adv. Appl.
    Hideo Kubo; Masahito Ohta, On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations
    BirkhäuserVerlag, 2005, 159, 113-211, [Contributor]
  • Dispersive Nonlinear Problems in Mathematical Physics
    KUBO Hideo, On point-wise decay estimates for the wave equation and their applications
    2004, 123-148, [Contributor]
■ Lectures, oral presentations, etc.
  • On the Vanishing Viscosity Approach to the Eikonal Equation via Neural Network
    Hideo Kubo
    Cosmology Building, National Taiwan University, 19 Jan. 2026, English
    [Invited]
  • Nonlinear wave equations with nonhomogeneous boundary condition in 1D
    久保英夫
    第13回弘前非線形方程式研究集会, 15 Nov. 2025, Invited oral presentation
    13 Nov. 2025 - 15 Nov. 2025, 40331129, [Invited]
  • Nonlinear wave equations on the half-line with nonlinear boundary condition
    Hideo Kubo
    The first Japan-Spain forum on PDEs and their applications, 26 Aug. 2025, English, Invited oral presentation
    25 Aug. 2025 - 29 Aug. 2025, 40331129, [Invited]
  • 半直線上の非線型境界条件を伴う波動方程式の最大存在時刻について
    久保英夫
    研究集会「微分方程式とその周辺」＠野田, 25 Jul. 2025, Japanese, Invited oral presentation
    40331129, [Invited]
  • Nonlinear wave equations on the half-line with nonlinear boundary condition
    Hideo Kubo
    International Workshop on Nonlinear Hyperbolic PDEs and related topics, 11 Jul. 2025, English, Invited oral presentation
    [Invited]
  • 幾何光学近似のニューラルネットについて
    久保英夫
    室蘭非線形波動方程式セミナー, 28 Jun. 2025, Japanese, Oral presentation
    50558039
  • Global existence for nonlinear wave equation with a singular potential
    久保英夫
    One day workshop on hyperbolic and dispersive PDEs, 22 Nov. 2024, Invited oral presentation
    40331129, [Invited]
  • Global existence for nonlinear wave equations with an inverse square potential
    久保 英夫
    The 21st Linear and Nonlinear Waves, 07 Nov. 2024, English, Invited oral presentation
    40331129, [Invited]
  • Global existence for nonlinear wave equations perturbed by the inverse-square potential below the Rellich constant
    久保英夫
    第18回浜松偏微分方程式研究集会, 26 Dec. 2023, Invited oral presentation
  • Global existence and blow-up for nonlinear wave equations with inverse-square potential
    Hideo Kubo
    The 24th Northeastern Symposium on Mathematical Analysis, 21 Feb. 2023, English, Invited oral presentation
    20 Feb. 2023 - 21 Feb. 2023, Sendai, Japan, [Invited], [International presentation]
  • 重み付きRellich 型不等式とその応用
    Hideo Kubo
    非線型偏微分方程式と走化性, 29 Nov. 2022, Japanese, Invited oral presentation
    29 Nov. 2022 - 01 Dec. 2022, 北九州市, Japan, [Invited], [Domestic Conference]
  • 逆二乗冪型ポテンシャルを伴う非線型波動方程式の解析 (PartⅡ)
    Hideo Kubo
    第43回発展方程式若手セミナー, 06 Sep. 2022, Japanese, Invited oral presentation
    05 Sep. 2022 - 07 Sep. 2022, [Invited]
  • 逆二乗冪型ポテンシャルを伴う非線型波動方程式の解析 (PartⅠ)
    Hideo Kubo
    第43回発展方程式若手セミナー, 05 Sep. 2022, Japanese, Invited oral presentation
    05 Sep. 2022 - 07 Sep. 2022, [Invited]
  • On the Rellich type inequality for Schrödinger operators with potential of inverse-square type
    Hideo Kubo
    Mathematical Analysis of Nonlinear Dispersive and Wave Equations, 25 Aug. 2022, English, Invited oral presentation
    24 Aug. 2022 - 26 Aug. 2022, Tokyo, Japan, [Invited], [International presentation]
  • Global existence for semilinear wave equations with potential of inverse-square type
    Hideo Kubo
    応用解析研究会, 23 Jul. 2022, English, Invited oral presentation
    23 Jul. 2022 - 23 Jul. 2022, 東京都, Japan, [Invited]
  • On the nonlinear wave equation with lower order terms
    Hideo Kubo
    Seminar of Applications of Differential Equations in Sciences, 22 Dec. 2021, English
    22 Dec. 2021 - 22 Dec. 2021, [Invited]
  • 低階項を伴う非線型波動方程式の初期値問題について
    久保英夫
    東京大学解析学火曜セミナー, 16 Nov. 2021, 東京大学, Japanese
    東京都, Japan, [Invited]
  • On the effect of slowly decreasing initial data for nonlinear wave equations with damping and potential in the scaling critical regime
    Hideo Kubo
    13th ISAAC Congress 2021, 03 Aug. 2021, English
    02 Aug. 2021 - 06 Aug. 2021, Ghent University, Belgium, [Invited], [International presentation]
  • On the semilinear wave equation with lower order terms
    久保英夫
    第37回 九州における偏微分方程式研究集会, 27 Jan. 2020
    [Invited]
  • 非線型波動方程式に対する幾何学的および双対的アプローチ (Part I)
    久保英夫
    第9回室蘭非線形解析研究会, 11 Jan. 2020
    [Invited]
  • Bio-inspired mathematical model of an effective integration of information
    KUBO Hideo
    第80回応用物理学会秋季学術講演会, 21 Sep. 2019, 公益社団法人 応用物理学会
    札幌市, [Invited], [International presentation]
  • Asymptotic behavior for the nonlinear damped wave equation with a positive potential
    KUBO Hideo
    信州大学偏微分方程式研究集会, 28 Jun. 2019, 信州大学
    松本市, [Invited]
  • ルールダイナミクスの適応性について
    KUBO Hideo
    On the activation of adaptive filters by the self-organization, 23 May 2019
  • Critical exponent for nonlinear damped wave equations with non-negative potential in 3D
    KUBO Hideo
    偏微分方程式セミナー, 26 Apr. 2019, 北海道大学
    北海道大学
  • 波動方程式に対する重み付きエネルギー評価とその周辺
    久保 英夫
    感応寺山セミナー2019, 19 Jan. 2019
    [Domestic Conference]
  • On the metric perturbation for semilinear wave equations
    KUBO Hideo
    SEMINARIO DI EQUAZIONI ALLE DERIVATE PARZIALI, 13 Dec. 2018, Università di Pisa, English
    Pisa, [Invited], [International presentation]
  • Global existence for nonlinear damped wave equations with a potential
    KUBO Hideo
    第１４回非線型の諸問題, 11 Sep. 2018
    [Invited]
  • Remark on Kolmogorov's superposition theorem
    KUBO Hideo
    RIMS共同研究「Mathematical Analysis of Self-Organization with Constraints」, 16 May 2018
  • Global existence for nonlinear damped wave equations with potential
    KUBO Hideo
    Zhejiang-Hokudai Workshop, 28 Mar. 2018
    [Invited], [International presentation]
  • On the exterior problem for systems of nonlinear wave equations with multiple speeds
    KUBO Hideo
    Workshop on Nonlinear Wave Equations, Apr. 2017, Fudan University
  • Asymptotic behavior of solutions to quasilinear wave equations with dissipative structure
    KUBO Hideo
    7th Euro-Japanese Workshop on Blow-up, Sep. 2016, The Mathematical Research and Conference Center
    Będlewo
  • On the local smoothing for the Dirac equation
    KUBO Hideo
    10th International ISAAC Congress, Aug. 2015, University of Macau
  • On the exterior problem for the wave equation with critical nonlinearity in 2D
    KUBO Hideo
    Analysis of Relativistic and Non-Relativistic models in Quantum Mechanics, Apr. 2014, University of Roma
  • On the null condition for nonlinear massless Dirac Equations in 3D
    KUBO Hideo
    Fourier Analysis and Pseudo-Differential Operators, Jun. 2012, Aalto University
  • Generalized wave operator for a system of nonlinear wave equations
    KUBO Hideo
    7th International ISAAC Congress, Jul. 2009, Imperial College London
  • Lifespan for nonlinear wave equations in an exterior domain
    KUBO Hideo
    SEMINARIE, Analyse numeric et E.D.P., Mar. 2009, Universite Paris-Sud
  • Large time behavior of solutions to semilinear wave equations with dispersive structure
    KUBO Hideo
    FRG/JAMI workshop “Nonlinear Dispersive Equations", Mar. 2007, Johns Hopkins University
  • Global and almost global existence for wave equations on unbounded domains
    KUBO Hideo
    6'eme Conf'erence Internationale AIMS, “Systemes Dynamiques, Equations Differentielles et Applications", Jun. 2006, Universite de Poitiers
  • 非線形波動方程式に対する散乱作用素の一つの構成法
    久保 英夫
    ENCOUNTER with MATHEMATICS “第31回スペクトル・散乱理論", Oct. 2004, 中央大学
  • 波動方程式の解の時空評価と非線型摂動への応用
    久保 英夫
    日本数学会函数方程式論特別講演, Sep. 2003, 千葉大学
  • On the small data global existence and scattering for systems of semilinear wave equations
    KUBO Hideo
    Hyperbolic Problems and Related Topics, Sep. 2002
    Cortona
  • Global existence to nonlinear wave equations with a potential in three dimensions
    KUBO Hideo
    微分方程式の総合的研究, Dec. 2000, 東京大学
  • Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition
    KUBO Hideo
    微分方程式の総合的研究, Dec. 1997, 大阪大学
■ Syllabus
  • ALP特別科目, 2024年, 修士課程, 大学院共通科目
  • 数学交流探究, 2024年, 修士課程, 理学院
  • 数理解析学講義, 2024年, 修士課程, 理学院
  • 大学院共通授業科目（一般科目）：自然科学・応用科学, 2024年, 修士課程, 大学院共通科目
  • 大学院共通授業科目（一般科目）：自然科学・応用科学, 2024年, 修士課程, 大学院共通科目
  • 大学院共通授業科目（一般科目）：自然科学・応用科学, 2024年, 修士課程, 大学院共通科目
  • 大学院共通授業科目（一般科目）：自然科学・応用科学, 2024年, 修士課程, 大学院共通科目
  • 微分積分学Ⅰ, 2024年, 学士課程, 全学教育
  • 微分積分学Ⅱ, 2024年, 学士課程, 全学教育
  • 数理解析学続論, 2024年, 学士課程, 理学部
  • 統計学, 2024年, 学士課程, 理学部
■ Affiliated academic society
  • 日本数学会
  • 日本数学会
■ Research Themes
  • データ駆動・モデル駆動の融合による神経ダイナミクス推定と脳機能制御の精緻化
    科学研究費助成事業
    08 Sep. 2023 - 31 Mar. 2029
    大森 敏明; 井上 広明; 久保 英夫
    本国際共同研究の初年度目にあたる令和５年度は，モデル駆動型手法とデータ駆動型手法の融合により，神経ダイナミクスの推定を精緻に行うための手法を構築した．神経細胞の時空間ダイナミクスを考慮したモデル駆動型アプローチとマルコフ連鎖モンテカルロ法に基づくデータ駆動型アプローチを融合することにより，神経システムの時空間ダイナミクスを精緻に推定するための手法を構成した．さらに，神経システムの数理モデルを考慮した非線形ダイナミクス制御を実現するアルゴリズムを構築し，入力強度に制約が存在する状況下でも，精緻な推定と制御を同時に実現する手法を構成した．加えて，計測データに基づいて，神経システムが受け取る入力の空間的構造を推定するアルゴリズムを構築した．グラフ構造を考慮した多次元自己回帰モデルを構成することで，入力時空間構造の精緻な推定を実現した．計測時系列データから背後に存在する動的システムの支配方程式を推定するために，ハミルトニアンのスパース表現に基づいた推定アルゴリズムを構成した．ハミルトニアンのスパース表現を導入することで，正準方程式における共通なスパース係数を導入するとともに，エネルギー保存則を考慮した推定アルゴリズムを構成し，時系列データからの動的システムの精緻な推定を実現した．以上の研究成果は，国内外の英文学術雑誌や国際学会論文として公表するとともに，関連研究者からの注目を受け，非線形ダイナミクス研究に関連する国際学会からの受賞を受けた．
    日本学術振興会, 国際共同研究加速基金(海外連携研究), 神戸大学, 23KK0184
  • 非線形消散波動方程式の一般論の構築と宇宙論および流体力学への応用
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
    Apr. 2022 - Mar. 2027
    高村 博之; 若杉 勇太; 加藤 正和; 佐々木 多希子; 久保 英夫; 津田谷 公利; 若狭 恭平
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), Tohoku University, Coinvestigator, 22H00097
  • 強双曲型方程式において弱零条件の果たす役割の解明
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    Apr. 2019 - Mar. 2024
    久保 英夫; 加藤 正和; 津田谷 公利; 若狭 恭平; Yordanov Borislav
    本研究の目的は、アインシュタイン方程式をプロトタイプとする強双曲型方程式に対する
    非線型摂動について、その安定性を弱零条件として特徴付けることである。その目的を達成するために、当該年度においては、アインシュタイン方程式を初期値問題として扱うための枠組みついての検討を詳細に亘って行った。具体的には、アインシュタイン方程式を扱う座標系を時間的座標軸が常に時間的であるように選ぶことによって得られる3+1形式に着目した。この定式化は数値相対論の分野で標準的に用いられているものである。まず、時空を空間的超平面によってスライスし、ラプス関数とシフトベクトルにより座標系を張る。アインシュタイン方程式の共変性に由来するゲージに関する自由度により、この様な座標系を採用しても一般性を失うことはない。この座標系においてローレンツ計量の3+1分解を行い、この分解に従ってアインシュタイン方程式を書き下すと、時間に依存しない拘束条件（ハミルトン拘束条件、運動量拘束条件）と時間発展する空間的超曲面の外的曲率に関する双曲型の方程式が得られる。これらの方程式系はADM形式と呼ばれるが、時間発展する方程式を導く際に、アインシュタイン・テンソルを表に出さず、リッチ・テンソルで表示されたアインシュタイン方程式を用いると数学的に扱いやすくなることが知られている。しかし、このADM形式において得られる方程式系は弱双曲型であり、初期値の微小摂動に関して時間大域的な安定性に問題があった。その困難を克服するために導入されたのがBSSN形式であり、実際、方程式系は強双曲型となり、アダマールの意味で適切となる。こうした理由から、我々はアインシュタイン方程式のBSSN形式を解析の対象とした。
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, Principal investigator, 19H01795
  • New development of mathematical theory of turbulence by collaboration of the nonlinear analysis and computational fluid dynamics
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
    May 2016 - Mar. 2021
    小薗 英雄; 三浦 英之; 久保 英夫; 木村 芳文; 芳松 克則; 前川 泰則; 隠居 良行; 金田 行雄; 小池 茂昭
    1. 尺度不変な斉次Besov空間における定常Navier-Stokes方程式の解の存在と正則性について
    n次元空間において，与えられた外力$f \in \dot B^{-3+\frac np}_{p, q}$ が十分小さければ，$u \in B^{-1+\frac np}_{p, q}$なる定常Navier-Stokes 方程式の解$u$ が一意的に存在することを証明した．ただし，1 ≦ p < ≦, 1 ≦ q ≦ ∞ である．応用として，定常Navier-Stokes 方程式に対する自己相似解が得られる．証明方法は，斉次Besov 空間$\dot B^s_{p, q}$, s>0 におけるHoelder型Leibnitz 規則と，n/p-s を指標とする埋め込み定理である．尚，鶴見により，仮定 1 ≦p < n かつ s>0 は最良であることが明らかにされた．
    2. Navier-Stoke流の影響下におけるKeller-Segel方程式系に対する時間大域的解の存在及び有限時間爆発の指標
    全平面領域における細胞性粘菌の密度$n$が，速度場 u を持つNavier-Stoke方程式に従う非圧縮性粘性流体の影響下にある場合を記述するKeller-Segel方程式系を，尺度不変な関数空間で考察した．まず，初期値$n_0 \in L^1(R^2)$, $u_0 \in L^2(R^2)$ が十分小さければ，時間大域的な古典解n, uが一意的に存在することを証明した．手法は線形熱半群の L^p-L^q 型評価とその摂動による．更に解が有限時刻で爆発する指標を，u_0 に何ら仮定を課すことなくn_0 のL^1における大きさで表現した．この指標は流体の影響がない場合の$\|n_0\|_{L^1(R^2)}$ の閾値 8π を含むものである．また爆発時刻 T における挙動を考察した．
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), Waseda University, Coinvestigator, 16H06339
  • Toward the integrated dynamics that connect evolutionary economics and engineering
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    Jul. 2016 - Mar. 2019
    Kubo Hideo; NISHIBE makoto
    Our personal behavior is conducted by the custom and routine of ourselves based on our experience and/or success stories. But in some cases, we are forced to change the usual strategy due to the change of the external circumstances. We studied such an adaptation to the stimuli from the outside world should not be designed by the top-down mechanism but by the bottom-up mechanism, in the framework of Mathematical analysis. In particular, we analyzed the immediate adaptation by following the way of information processing used by insects.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, Principal investigator, 16KT0015
  • Advanced Analysis on Evolving Patterns in Nonlinear Phenomena Driven by Singular Structure
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
    May 2014 - Mar. 2019
    GIGA Yoshikazu
    We prove the existence and the uniqueness of a solution and clarify its behavior for evolution equations mainly nonlinear diffusion equations describing evolution of patterns and shapes like crystal growth phenomena. We introduce new notions of a solution which allows shape with singularities for equations having singular structure. We thus establish foundation of mathematical analysis which easily describes real phenomena. Based on these fundamental results, we are able to numerically calculate phenomena which had been difficult to calculate, for example, phenomena of colliding spirals on surfaces of crystals.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), The University of Tokyo, Coinvestigator, 26220702
  • An investigation of symmetries in the geometric structure and existence of global solutions to nonlinear dispersive wave equations
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    Apr. 2013 - Mar. 2018
    Takaoka Hideo; KUBO Hideo; NAKANISHI Kenji; TSUGAWA Kotaro
    In this study, I have developed the local and global well-posedness for the initial value problem related to the nonlinear Schrodinger equations in which dispersion effect and nonlinear interaction effect are incorporating. Using the Fourier analysis, I separated the solution into two parts; non-resonant and resonant oscillation parts, which have different in nature and distinguish nonuniformity part of solutions. For the nonlinear Schrodinger equations both with derivative in nonlinearities and on a sphere domain, I improved the local well-posedness for large function spaces. Moreover, I showed that there exists exchange of energy between Fourier modes. In the research process, I observed the estimation of energy exchange between different Fourier modes, due to the contribution in the nonlinear interaction.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Coinvestigator, 25287022
  • Biomimetics based on the functional structure and the formation process of organisms
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)
    Jun. 2012 - Mar. 2017
    HARIYAMA Takahiko
    In order to embody the material design based on the surface structure of organisms and to develop the energy saving production process, we focused on the moth eye structures and the structural colors of several organisms; 1. Fabrication of high brightness surface structure by self-organizing method, 2. Observation of morphogenesis of organisms for industrialization, 3. To discover the meaning of the organism's "not precise structure but precise function", we organized research teams of different fields including mathematics, physics, biology, chemistry and engineering, and clarified their functions.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area), Hamamatsu University School of Medicine, Coinvestigator, 24120004
  • Mathematical Theory of turbulence by the method of modern analysis and computational science
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
    May 2012 - Mar. 2017
    KOZONO HIDEO; OZAWA Tohru
    The challenging problem on global well-posedness of the Navier-Stokes equations had been so fully investigated that several remarkable results are obtained. Furthermore, our DNS of the uniformly isotropic turbulence is still by far the larger computational performance so that we could deal with the turbulent fluid with the high Reynolds number without any error of the experiment and indeterminacy. Our study has been based on the DNS of such a world highest standard and we could succeed to overcome difficulty of turbulence with the high Reynolds number. In this way, our research projects have developed the modern mathematical analysis, the applied mathematics, computational science and hydrodynamics and hopefully will lead the relevant subjects to the world-wide level.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), Waseda University, Coinvestigator, 24224003
  • Global behavior for nonlinear wave
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    Apr. 2012 - Mar. 2016
    Kubo Hideo; KATAYAMA Soichiro; TAKAMURA Hiroyuki; HOSHIGA Akira; NAKAMURA Makoto; DOI Kazuyuki
    Our research is concerned with equations which describe the way of propagation of waves. More precisely, we study the nonlinear effect produced by the interaction among waves, as well as the effect coming from the structure of the space-time in which the wave exists. For instance, when there exists an obstacle in the space, we are able to show that the way of propagation of waves are similar to that for the case where there exists no obstacle. In particular, it is a big progress to solve this kind of problem in two space dimensional case.
    Moreover, we also study the way of propagation of waves in the space-time equipped with the metric which describes the expanding universe. By considering the property of function which represents the wave in detail, we rigorously proved the following intuitive image: the wavelength of the waves become long, so that the waves are stabilized in such an expanding universe model.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, Principal investigator, 24340024
  • Quantum stochastic analysis - Transforms and spectral analysis
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    Apr. 2011 - Mar. 2015
    OBATA Nobuaki; FUKUIZUMI Reika; HASEGAWA Takehisa; SEGAWA Etsuo; KUBO Hideo; HIAI Fumio; SUZUKI Kanako
    For the development of quantum stochastic analysis we focused on 'quantum white noise calculus' from analytic aspect and 'spectral analysis of complex networks' from algebraic aspect. We aimed at the establishment of the mathematical fundamentals and the paradigm for collaborating with other research fields for applications. By means of quantum white noise calculus, the Bogoliubov transform and the Girsanov transform are characterized by the white noise differential equations of new types. A quantum probabilistic method is applied to the spectral analysis of digraphs such as Manhattan product. The phase transition of various dynamics on networks is studied in detail with the help of numerical computation. New statistical properties of quantum walks on graphs such as localization are obtained by generalizing the existing method of spectral analysis.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Tohoku University, Coinvestigator, 23340027
  • Analysis of differential equations on graphs.
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    2011 - 2013
    TRUSHIN Igor; KUBO Hideo; MOCHIZUKI Hiyoshi
    In this project we consider Schrodinger operators on noncompact graphs which consist of some infinite rays and compact part attached. Spectral and scattering problems on graphs arise as simplified models in mathematics, physics, chemistry and engineering when one considers the propagation of waves of different natures in thin, tube-like domains. We study scattering direct and inverse problems which are important in applied physics. (1)We treat an inverse scattering problem on a graph with an infinite ray and a loop joined at one point. Reconstruction procedure is presented.(2)We consider Schrodinger operators on noncompact star-shaped graphs including some finite rays. We show that our spectral representation formula provides the time dependent formulation of the scattering theory. The scattering operator is constructed in the configuration space, and then is related to the scattering matrix in the momentum space. Corresponding inverse scattering problem is investigated.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Tohoku University, Coinvestigator, 23540181
  • Theory of global well-posedness on the nonlinear partial differential equations
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
    2008 - 2012
    KOZONO Hideo; YANAGIDA Eiji; ISHIGE Kazuhiro; NAKAMURA Makoto; KUBO Hideo; KANEDA Yukio; ISHIHARA Takashi; YOSHIMATSU Katsunori; KAGEI Yoshiyuki; EI Shinichro
    We investigate the local existence of strong solutions and their blow-up within a finite time in arbitrary dimensional domains. The life-span of local solutions is characterized in terms of the L^1 and L^p-norms of the given initial data. Simultaneously, it is clarified that the total mass and the second momentum of the initial data together with the coefficient of the system of equations have a great influence on the blow-up phenomena. As an application, we prove that the blow-up solution either exhibits a definite blow-up rate determined by p, or oscillates in L^1 with the larger amplitude than the absolute constant. Furthermore, in multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), Coinvestigator, 20224013
  • On the limiting amplitude principle for the exterior problem of the wave equation
    Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
    2010 - 2011
    KUBO Hideo
    Large time behavior of solutions to the exterior problem whose boundary value is oscillating in time is considered. It can be expressed as a product of a time periodic function with the same period as the boundary value and the resonance of the correspo nding Helmholtz equation. For the radially symmetric case, the existence of the resonance is actually proved. In conclusion, the limiting amplitude principle for the exterior problem of the wave equation with a periodic boundary value was formulated.
    Japan Society for the Promotion of Science, Grant-in-Aid for Challenging Exploratory Research, Tohoku University, Principal investigator, 22654017
  • Research of global behavior of classical solutions for quasilinear Wave equations
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    2008 - 2010
    HOSHIGA Akira; KATAYAMA Soichiro; KUBO Hideo; KUROKAWA Yuki
    In this study, we succeeded in the classification of the sufficient conditions (null-conditions) for the global existence of the classical solutions to the system of nonlinear wave equations in 2 and 3 space dimensions, according to the type of the nonlinear terms (Null-form type, Non-resonance type and Nonlinear dissipation type). As development of the research, we also obtained precise evaluations to the lifespan of the classical solutions for the first order hyperbolic PDE systems with multiple propagation speeds and for the nonlinear elastic wave equations.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 20540206
  • Phase Space Analysis of Partial Differential Equations
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
    2007 - 2010
    NISHITANI Tatsuo; HAYASHI Nakao; DOI Shinichi; SUGIMOTO Mitsuru; SUNAGAWA Hideaki; KUBO Hideo; TAKUWA Hideki; UMEDA Tomio; IWASAKI Chisato; HOSHIRO Toshihiko; FUJIIE SETSURO; TOMITA Naohito
    Much progress has been achieved on linear hyperbolic Cauchy problem, on precise asymptotic behaviors of solutions to nonlinear dissipative and wave equations and on semi-classical resonances, by local and global phase space analysis, in deep cooperation with all research members through annual international meeting. We have also successfully supported young mathematicians to acquire the techniques of phase space analysis by annual instructive conference.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), Osaka University, Coinvestigator, 19204013
  • On study of evolution equations with hyperbolic properties
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    2007 - 2010
    HAYASHI Nakao; NISHITANI Tatsuo; DOI Shinichi; KUBO Hideo
    We studied nonlinear Schrodinger equations, nonlinear Klein-Gordon equations and their systems. Time decay and asymptotic behavior of solutions were shown. We applied these results to show existence of wave or modified wave operators. In the case of critical nonlinearities, it was shown that main terms of solutions can be represented through nonlinearities clearly.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Osaka University, Coinvestigator, 19340030
  • On the asymptotic behavior of solution to systems of nonlinear wave equations of long range type
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    2005 - 2007
    KUBO Hideo; HAYASHI Nakao; MATSUMURA Akitaka
    The aim of this research is to study the asymptotic behavior of wave functions perturbed by the influence of the nonlinearity and characterize such asymptotic behavior. It is known that if the influence of the nonlinearity is too strong, then the wave function diverges in a finite time. On the other hand, if the influence of the nonlinearity is weak, then the wave function exists globally in time and it tends to a wave function which is free from the nonlinear perturbation in the sense of the energy as time goes to infinity.
    In this research, we treat the intermediate case, namely, we are interested in the case where the perturbed wave function exists globally in time, but it does not tend to any free wave function as time goes to infinity. In order to consider such nonlinear perturbation, our first task is to find nonlinear wave equations which admit global in time solutions whose asymptotic behavior may differ from any solution to the corresponding homogeneous wave equations. Then the next step is to show that its asymptotic behavior is actually different from the free solution. As for these problems, we seemed to find several examples of such nonlinear perturbation. For some examples, the asymptotic behavior of the wave function is better compared with that of the free solution. On the other hand, it is worse than that of the free solution for the other examples. Such difference is determined by a quantity which is computed from the order and the coefficients of the nonlinearity.
    In the former case, the asymptotic profile is given by a second iterate of the free solution. On the other hand, in the latter case, the asymptotic profile is closely related to the radiation field for the free solution. We obtain a suitable ordinary differential equation whose solution gives the modification of the free radiation field.
    In conclusion, the nonlinear perturbation of long range type is complicated and contains a full of variety to produce different kinds of asymptotic behavior.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Osaka University, Principal investigator, 17540157
  • On study of partial differential equations describing natural phenomena
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
    2003 - 2005
    HAYASHI Nakao; NISHITANI Tatsuo; DOI Shin-ichi; MATSUMURA Akitaka; KUBO Hideo; SUGIMOTO Mitsuru
    1,P.I.Naumkin and I studied the Burgers equation with pumping and showed a existence in time of solutions and asymptotic behavior of solutions by using a suitable transformation and the structure of nonlinear term.
    2,E.I.Kaikina and I studied the KdV equations in a half line with 0 boundary value at the origin. Airy function is oscillating rapidly in the left hand side and decaying exponentially in the right hand side. We showed asymptotics of solutions to the KdV equation by making use of this property.
    3,E.I.Kaikina, P.I.Naumkin and I studied nonlinear complex dissipative equations with sub-critical nonlinearities and showed a solution is stable in the neighborhood of a self similar, solution.
    4,P.I.Naumkin, Shimomura, Tonegawa and I did a joint work on nonlinear Schredinger equations with cubic nonlinearities. It was known that there exists a modified wave operator under some geometric assumptions on the final data. We succeeded to remove a strong geometric assumption by finding a new way to get a second approximate solution of the problem.
    5,E.I.Kaikina, P.I.Naumkin and I studied nonlinear damped wave equations with super-critical or critical nonlinearities. In the previous works, it was known that a global existence theorem holds in space dimension is less than 5. We improved this result for any space dimension by using the weighted Sobolev spaces and estimates of solutions linear problem. Furthermore, in the critical case we showed asymptotics of solutions. The result implies the decay order in time of solutions is higher than that of solutions to linear problem. We obtained the results by using the method we found in the study of nonlinear dissipative equations
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), Osaka University, Coinvestigator, 15204009
  • Mathematical analysis of interface problems in mathematical physics
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    2002 - 2004
    SHIMIZU Senjo; SHIBATA Yoshihiro; KIKUCHI Koji; HOSHIGA Akira; ADACHI Shinji; NAKAJIMA Toru
    In this research, we consider the Stokes equation with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equation. We analyzed this problem by the following procedure : (1) Analysis of the resolvent problem (2) Generation of Analytic semigroups (3) L_p-L_q estimates
    (1)Obtained is the L_p estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in R^n. The result has been obtained by Grubb and Solonnikov by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from theirs. The core of my approach is to estimate the solutions in the whole space and half-space case. We apply the Fourier multiplier theorem to solution of the model problems.
    (2)First we introduce the Helmholtz decomposition. Then we delete pressure trem and reduce to the problem only including velocity vector. Then we generated analytic semigroup to this reduced Stokes equation.
    (3)We obtained local energy decay estimates and L_p-L_q estimates of the solutions to the Stokes equation with Neumann boudary condition. Comparing with the non-slip (Dirichlet) boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null net force at the boundary.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 14540171
  • 摂動型波動方程式に対する重みつき時空評価に関する研究
    Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)
    2002 - 2004
    久保 英夫
    本研究の目的は波動方程式においてその線型部分が摂動された方程式を解析し,その解の挙動と摂動項がない方程式の解のそれとの違いを調べることである.時間と空間がある意味で対等であるという波動方程式の性質から,その解の挙動は時間変数と空間変数の混在した形の減衰評価によって,より良く近似されると考えられる.そこで,重みつき時空評価がどのような形で摂動型波動方程式の解について成り立つか考察した.
    まず,ポテンシャル項による摂動のある場合に重みつき時空評価を摂動のない場合と同様な形で導いた.しかし,質量項がない場合にはポテンシャルが無限遠方で十分速く減衰しているという仮定が必要であり,他方,質量項のある場合にはポテンシャルの減衰をそれ程必要としない代わりに最終的な評価は微分の損失を含んでいる.前者の評価式は更に非線型問題への応用が可能である.この様な評価を導くために散乱理論・フーリエ積分作用素・補間空間論などの理論を用いた.
    また,非線型項による摂動による影響が重みつき時空評価にどのように影響するかについても調べた.非線形項の次数が高ければ,小さな解に対して摂動のない解が満たすのと同様の重みつき時空評価が得られた.このような評価式は,伝播速度の異なる非線型波動方程式系を解析するのにも有効である.更に,空間2次元の問題を扱うとき,時空評価からルベーグ空間における評価を導くことによって,より広いクラスの非線型項に対して時間大域解の存在を示すことが出来ることが分かった.
    Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B), Principal investigator, 14740114
  • Analysis of gradient flow equations and Lagrange equations of action integrals associated to quasiconvex functionals
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    2002 - 2003
    KIKUCHI Koji; KUBO Hideo; SHIMIZU Senjo; NEGORO Akira; NAKAJIMA Toru; HOSHIGA Akira; OHTA Masahito
    This research was projected in order to investigate the following problems. 1.Constructing gradient flows associated to typical quasiconvex functionals, 2.Study in Lagrange equations of action integrals associated to typical quasiconvex functionals, 3.Discovering phenomena that show differences between convex and quasiconvex functions. During the term of the project the head investigator, Kikuchi, attended various conferences and discussed with specialists in related research areas. In the second year Workshop on spectral theory and differential operators was held at Fudan University, Shanghai, China, and the head investigator attended this conference, anounced his recent result and gathered information. Other investigators also attended various conferences held in Japan or abroad and gathered recent information. Thereby following research results are obtained. The most progresses are obtained in Problem 2. Linear application is investigated for a Lagrange equation of an action integrals associated to a functional that corresponds a value of the integral of F(Du(x)) for a function u, and several results are obtained in case that F is quasiconvex and linear growth. Before obtaining this result, it is obtained for the same equation that a sequence of approximate solutions to this equation constructed by Rothe's method converges to a function and that, if it satisfies the energy conservation law, it is a weak solution in the space of BV functions. This is already established for convex cases, and now it is successfully established for quasiconvex cases. Related to Problem 3, the problem requires a different observation from that in convex cases. So far, energy inequality is obtained by the use of the convexity of the functional, and hence this method is not available in quasiconvex cases. Instead our constructiong approximate solutions elementwisely makes it possible to obtain energy inequality. This seems to be a large difference between convex and quasiconvex functions. In research related to Problem 1, although constructing a gradient flow is not complete, it is sucseeded to find an identity in the process of constructing approximate solutions, which should be a key for our destination.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 14540202
  • 非線型波動におけるStrauss予想のある一般化
    Grants-in-Aid for Scientific Research Grant-in-Aid for Encouragement of Young Scientists (A)
    2000 - 2001
    久保 英夫
    本研究では、べき乗型の非線型項をもつ波動方程式の初期値問題を一般次元において扱った。台コンパクトな初期値については、その大きさがある意味で十分小さければ、ある臨界指数が存在して、非線型項の原点近傍でのオーダーがその臨界指数よりも真に大きいとき、時間大域的に弱解が存在することが知られていた。また、初期値が球対称の場合には、その無限遠方での減衰度に関する臨界オーダーのあることが知られていた。ここでは、球対称とは限らない一般の初期値に対して大域可解性を示すことを目標とし、ほぼ満足のいく結果が得られた。
    証明の要点は次の2点である。一つは、斉次波動方程式に対する初期値問題の解を適切な重みつきルベーグ空間で評価できたこと。それには、初期値が属する空間として、通常の微分作用素だけではなく、ローレンツ群に付随するリー代数を表現するベクトル場も加えた微分作用素から生成される重みつきソボレフ空間を採用したことが決め手となった。もう一つは、非斉次波動方程式に対する初期値問題の解について、非斉次項が各時刻において台コンパクトであるという仮定のもとに得られていた評価を一般の場合に拡張したことである。そのために、スケーリングの議論を適用し、非斉次項の空間無限遠方での適当な可積分性の仮定のもと、必要な不等式を導くことができた。
    以上の準備のもと、よく知られた手順に従って、小さな初期値に対して時間大域解の存在を証明された。結果として、初期値が予想される臨界オーダーより真に速く減衰していればよいことを、初期値の属する空間の性質から結論することができる。
    Japan Society for the Promotion of Science, Grant-in-Aid for Encouragement of Young Scientists (A), Shizuoka University, Principal investigator, 12740105
  • Research in evolution equations related to variational problems
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    2000 - 2001
    KIKUCHI Koji; KUBO Hideo; SHIMIZU Senjo; NEGORO Akira; OHTA Masahito; HOSHIGA Akira
    This research was projected in order to investigate the following problems. 1. Constructing gradient flows for various variational problems in, for example, nonlinear elasticity, 2. Bifurcation phenomena for gradient flow equations, 3. Hyperbolic equations related to deformation of elasticity and to area functional, 4. Application of the method of discrete Morse semiflow to the theory of Schrodinger equations, 5. Relation between blowup solutions and the method of discrete Morse semiflow. In the first year of this project World Congress of Nonlinear Analysts which is held once in each four years was held and hence the head investigator, Kikuchi, and another investigator, Ohta, attended this congress and gathered some recent information related to this project. In the second year Czechoslovak International Conference on Differential Equations and Their Applications was held and the head investigator attended this conference, anounced his recent result and gathered information. Besides each investigators attended various conferences held in Japan or abroad, announced each results and gattered recent information. Thereby following research results have been obtained. The most progresses are obtained in problems 1 and 3. The result related to 1 is that a gradient flow can be consructed when a quasiconvex functional satisfies some coersiveness condition. Furthermore, though the form of equation is restrictive, it turns out that a gradient flow for some quasiconvex functional can be constructed even if it does not satisfy such a coersiveness condition. The result related to 3 is that Dirichle condition for the equation of motion of vibrating membrane should be weaker than the usual weak formulation (that the trace vanishes). This result is obtined by applying a result in direct variational method to the theory of evolution equations, what is the most feature of this research project. Some facts related to Problem 4 are also obtained. It is confident that some new theories related to 2 and 5 will also be developed. But by now frames of these works have not yet been obtained. It should be expected in the future.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 12640205
  • Research on properties of Markov processes governed by the pseudo-differential operators with variable orders and application of the m to nonlinear analysis
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    1998 - 1999
    NEGORO Akira; KUBO Hideo; KIKUCHI Koji; TAKANO Masaru
    As is well known, under suitable conditions, it has been shown that there exist pure jump type Markov processes governed by Levy. generating operators with degenerate Levy mesures. So we would like to know what conditions these Markov processes have their transition densities under. Recently, by using MALLIAVIN calculus, Kunita has constructed transition densities of these Markov proceses in some class. So, we tried to adapt the pseudo-differential operators theory for this problem and restricted our study to the case that the supports of Levy measures degenerated into mutualy independent d lines for each x in RィイD1dィエD1. Cosequently, we have got that Markov processes governed the following generators, L have transition densities. The L is
    <>
    where θィイD2jィエD2(x) (j = 1, 2,…, d) are smooth RィイD1dィエD1-valued functions with bounded derivatives on RィイD1dィエD1 and satisfy |θィイD2jィエD2(x)|=1(j = 1, 2,…, d). Putting Θ(x)=(θィイD21ィエD2(x), θィイD22ィエD2(x), …, θィイD2dィエD2(x)), we assume that the eigenvalues of Θ(x)*Θ(x) are unifomly bounded to the below. And also, α is a constant satisfying 1 < α < 2 and nィイD2jィエD2(x,y) (j = 1,…, d) are smooth funcutions with bounded derivatives satisfying usual coditions. Now, we are rounding off the above work. We regret to say that we were able to have no result about the relation between nolinear differential operators and stochastic processes. But while we were studing this problem, we had the following results.
    (1) A one dimensional hyperbolic equation uィイD2ttィエD2 - uィイD2xxィエD2 = 0 is treated under a free boundary condition uィイD32(/)XィエD3-uィイD32(/)tィエD3=QィイD12ィエD1. The existance and the uniqueness of a classical solution is established loccaly.
    (2) A weak solution to some forth order nonlinear parabolic equation is constructed by the method of time semidisceretization. A technique of geometoric measure theory is employed in order to obtain to obtain the convergence of the nonlinear terms.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shizuoka University, Coinvestigator, 10640159
  • 非線型波動方程式に関する研究
    1996
    Competitive research funding
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