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KOHSAKA Yoshihito
Graduate School of Maritime Sciences / Department of Maritime Sciences
Professor

Researcher basic information

■ Research Areas
  • Natural sciences / Mathematical analysis
■ Committee History
  • 2010 - 2011, 日本数学会, 評議員会, 日本数学会
  • 2010 - 2011, 日本数学会, 「数学通信」編集委員会, 日本数学会

Research activity information

■ Paper
  • Katsuyuki Ishii, Yoshihito Kohsaka, Nobuhito Miyake, Koya Sakakibara
    We propose a thresholding algorithm for Willmore-type flows in \mathbb{R}^{N} . This algorithm is constructed based on the asymptotic expansion of the solution to the initial value problem for a fourth-order linear parabolic partial differential equation whose initial data is the indicator function on the compact set \Omega_{0} . The main results of this paper demonstrate that the boundary \partial\Omega(t) of the new set \Omega(t) , generated by our algorithm, is included in O(t) -neighborhood of \partial\Omega_{0} for small t>0 and that the normal velocity from \partial\Omega_{0} to \partial\Omega(t) is nearly equal to the L^{2} -gradient of Willmore-type energy for small t>0 . Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our thresholding algorithm.
    European Mathematical Society - EMS - Publishing House GmbH, Nov. 2024, Interfaces and Free Boundaries, Mathematical Analysis, Computation and Applications, 27(2) (2), 235 - 283, English
    [Refereed]
    Scientific journal
    Link to Kobe Univ. Repository (Kernel)

  • Takashi Kagaya, Yoshihito Kohsaka
    Dec. 2020, Advanced Studies in Pure Mathematics, 85, 227 - 238, English
    [Refereed]
    International conference proceedings

  • Takashi Kagaya, Yoshihito Kohsaka
    Jan. 2020, Archive for Rational Mechanics and Analysis, 235(1) (1), 471 - 516, English
    [Refereed]
    Scientific journal

  • On the criteria for the stability of unduloids
    Yoshihito Kohsaka
    RIMS, 2017, RIMS Kokyuroku Bessatsu, B63, 167 - 192, English
    [Refereed]
    International conference proceedings

  • Yoshihito Kohsaka
    2016, GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE'S, 176, 121 - 148, English
    [Refereed]
    International conference proceedings

  • Yan-Yu Chen, Yoshihito Kohsaka, Hirokazu Ninomiya
    May 2014, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 19(3) (3), 697 - 714, English
    [Refereed]
    Scientific journal

  • Daniel Depner, Harald Garcke, Yoshihito Kohsaka
    Jan. 2014, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 211(1) (1), 301 - 334, English
    [Refereed]
    Scientific journal

  • SURFACE DIFFUSION WITH TRIPLE JUNCTIONS: A STABILITY CRITERION FOR STATIONARY SOLUTIONS
    Harald Garcke, Kazuo Ito, Yoshihito Kohsaka
    May 2010, ADVANCES IN DIFFERENTIAL EQUATIONS, 15(5-6) (5-6), 437 - 472, English
    [Refereed]
    Scientific journal

  • Nonlinear stability of stationary solutions for curvature flow with triple junction
    Harald Garcke, Yoshihito Kohsaka, Daniel Sevcovic
    Nov. 2009, HOKKAIDO MATHEMATICAL JOURNAL, 38(4) (4), 721 - 769, English
    [Refereed]
    Scientific journal

  • Garcke Harald, Ito Kazuo, Kohsaka Yoshihito
    2009, Banach Center Publ., 86, 83 - 101, English
    [Refereed]
    International conference proceedings

  • Harald Garcke, Kazuo Ito, Yoshihito Kohsaka
    2008, SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 40(2) (2), 491 - 515, English
    [Refereed]
    Scientific journal

  • ON THE EXISTENCE OF SOLUTIONS OF THE HELFRICH FLOW AND ITS CENTER MANIFOLD NEAR SPHERES
    Yoshihito Kohsaka, Takeyuki Nagasawa
    Feb. 2006, DIFFERENTIAL AND INTEGRAL EQUATIONS, 19(2) (2), 121 - 142, English
    [Refereed]
    Scientific journal

  • H Garcke, K Ito, Y Kohsaka
    2005, SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 36(4) (4), 1031 - 1056, English
    [Refereed]
    Scientific journal

  • On a two-point free boundary problem for a quasilinear parabolic equation
    YL Chang, JS Guo, Y Kohsaka
    Jun. 2003, ASYMPTOTIC ANALYSIS, 34(3-4) (3-4), 333 - 358, English
    [Refereed]
    Scientific journal

  • Spiral solutions for a weakly anisotropic curvature flow equation
    Giga Yoshikazu, Ishimura Naoyuki, Kohsaka Yoshihito
    2002, Advances in Mathematical Sciences and Applications, 12(1) (1), 393 - 408, English
    [Refereed]
    Scientific journal

  • Y Kohsaka
    Sep. 2001, NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 45(7) (7), 865 - 894, English
    [Refereed]
    Scientific journal

  • Three phase boundary motion by surface diffusion in triangular domain
    Ito Kazuo, Kohsaka Yoshihito
    2001, Advances in Mathematical Sciences and Applications, 11(2) (2), 753 - 779, English
    [Refereed]
    Scientific journal

  • Kazuo Ito, Yoshihito Kohsaka
    European Mathematical Society Publishing House, 2001, Interfaces and Free Boundaries, 3(1) (1), 45 - 80, English
    [Refereed]
    Scientific journal

■ MISC
  • Motion of phase boundaries by geometric evolution equations
    2010

  • On existence of a solution for curvature flow with triple junction
    2009

  • Phase boundary motion by surface diffusion with triple junction
    2008

  • 表面拡散流方程式の時間大域解の存在と定常解の安定性について
    2007

  • 表面拡散流方程式による相境界運動の定常解の安定性について
    2007, 「熊本大学応用解析セミナー」報告集, 43 - 47

  • 幾何学的時間発展方程式の時間大域解の存在と定常解の安定性について
    2007

  • Stability analysis of phase boundary motion by surface diffusion with triple junction
    2007

  • 表面拡散流方程式による３相境界運動の数学的解析について
    2006

  • Stability of stationary solutions for surface diffusion flow with boundary conditions
    2006

  • Stability analysis of phase boundary motion by surface diffusion in a bounded domain
    2006

  • 表面拡散流方程式による3相境界運動の定常解の線形安定性について
    2005, 「偏微分方程式と現象」報告集, 98 - 108

  • Linearized stability analysis of three phase boundary motion by surface diffusion in a bounded domain
    2005

  • 表面拡散流方程式による３相境界運動の定常解の線形安定性について
    2004

  • Linearized stability analysis of three phase boundary motion by surface diffusion
    2004

  • A criterion of linearized stability of stationary solutions for surface diffusion flow equation
    2004

  • Stability analysis of stationary solutions for surface diffusion flow equation
    2004, RIMS Kokyuroku, 1356, 27 - 43

  • Linearized stability analysis of three phase boundary motion by surface diffusion
    2004

  • A criterion of linearized stability of stationary solutions for surface diffusion flow equation
    2004

  • Stability analysis of stationary solutions for surface diffusion flow equation
    2003

  • On time-periodic solution for a weakly anisotropic curvature flow equation with driving force term
    2003, 数理解析研究所講究録, 1323, 13 - 26

  • On phase boundary motion by surface diffusion in a bounded domain
    2003

  • Linear stability analysis of stationary solutions for surface diffusion flow equation in a bounded domain
    2003, Proceedings of the Ryukoku Workshop 2003, 68 - 77

  • On time-periodic solution for a weakly anisotropic curvature flow equation with driving force term
    2003, RIMS Kokyuroku, 1323, 13 - 26

  • Stability analysis of stationary solutions for surface diffusion flow equation
    2003

  • On phase boundary motion by surface diffusion in a bounded domain
    2003

  • Linear stability analysis of stationary solutions for surface diffusion flow equation in a bounded domain
    2003, 68 - 77

  • Three phase boundary motions in triangular domain
    2002

  • Self-similar solutions of two-point free boundary problem for heat equation
    2002, RIMS Kokyuroku, 1258, 94 - 107

  • Three phase boundary motions in triangular domain
    2002

  • Some topics of three-phase boundary problem for a surface diffusion equation
    2001

  • On free boundary problem arising in the combustion theory
    2001

  • Three phase boundary motion by surface diffusion
    2001, RIMS Kokyuroku, 1210, 43 - 53

  • On free boundary problem arising in the combustion theory
    2001

  • On three-phase boundary motion for a fourth order model - symmetric case
    2000, Free Boundary Problems: Theory and Applications II, 207 - 221

  • Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary
    1999, RIMS Kokyuroku, 1076, 41 - 56

■ Lectures, oral presentations, etc.
  • 幾何学的発展方程式に対する閾値型近似アルゴリズム
    高坂良史
    第47回さいたま数理解析セミナー, Mar. 2026
    [Invited]

  • Willmore 流の閾値型近似アルゴリズムについて
    高坂良史
    微分方程式とその周辺＠野田, Jan. 2026
    [Invited]

  • On the existence of a self-similar solution to the Mullins' problem
    Yoshihito Kohsaka
    Matrix Research Program: Gradient Flows in Geometry and PDE, Jan. 2025, English
    [Invited]

  • Helfrich型エネルギーの勾配流に対する閾値型近似アルゴリズムについて
    高坂良史
    日本応用数理学会2024年度年会, Sep. 2024

  • A threshold-type algorithm to the gradient flow of the Canham-Helfrich functional
    Yoshihito Kohsaka
    Regularity and Singularity for Geometric PDE and related topics, Jan. 2024, English
    [Invited]
    Invited oral presentation

  • Canham-Helfrich汎関数の勾配流に対する閾値型近似
    高坂良史
    第73回南大阪応用数学セミナー, Nov. 2023, Japanese
    [Invited]
    Public discourse

  • Willmore流に対する閾値型近似アルゴリズムについて
    高坂良史
    2023年度秋季総合分科会 函数方程式論分科会, Sep. 2023
    [Invited]
    Invited oral presentation

  • 幾何学的発展方程式に対する進行波解について
    高坂良史
    「応用解析」研究会定例セミナー, Apr. 2023, Japanese
    [Invited]
    Public discourse

  • 4階幾何学的発展方程式による曲線の挙動解析について
    高坂良史
    北陸応用数理研究会2023, Mar. 2023, Japanese
    [Invited]
    Invited oral presentation

  • On traveling waves of geometric evolution equations with area con- straints
    Yoshihito Kohsaka
    The 40th Kyushu Symposium on Partial Differential Equations, Jan. 2023, English
    [Invited]
    Invited oral presentation

  • 体積保存型幾何学的発展方程式の進行波解について
    高坂良史
    非線型偏微分方程式と走化性, Nov. 2022, Japanese
    [Invited]
    Invited oral presentation

  • 4階幾何学的発展方程式について
    高坂良史
    さいたま数理解析セミナー, Nov. 2022, Japanese
    [Invited]
    Invited oral presentation

  • 表面拡散方程式の境界値問題に対する進行波解について
    高坂良史
    RIMS共同研究(公開型)偏微分方程式の臨界現象と正則性理論及び漸近解析, Jun. 2022, Japanese
    [Invited]
    Invited oral presentation

  • Stability of steady states for geometric evolution equations
    高坂良史
    第45回偏微分方程式論札幌シンポジウム, Aug. 2020, Japanese
    [Invited]
    Invited oral presentation

  • Stability of the steady states for evolving surfaces by surface diffusion
    Yoshihito Kohsaka
    Mini-symposium Nonlinear Geometric Partial Differential Equations, Feb. 2020, English
    [Invited]
    Invited oral presentation

  • 接触角境界条件をもつ表面拡散方程式に対する進行波解について
    高坂良史
    第31回さいたま数理解析セミナー, Jan. 2020, Japanese
    [Invited]
    Public discourse

  • On the traveling waves for surface diffusion of curves with constant contact angles
    Yoshihito Kohsaka
    Workshop on Elliptic & Parabolic PDEs 2019, Nov. 2019, English
    [Invited]
    Invited oral presentation

  • Stability of axisymmetric CMC surfaces as the steady states for the surface diffusion equation
    KOHSAKA YOSHIHITO
    7th International Conference on Mathematical Modeling in Physical Sciences, Aug. 2018, English, Moscow State University, International conference
    Oral presentation

  • On the dynamics for the volume-preserving geometric flows
    KOHSAKA YOSHIHITO
    The 11th MSJ-SI The Role of Metrics in the Theory of Partial Differential Equations, Jul. 2018, English, Hokkaido University, International conference
    [Invited]
    Invited oral presentation

  • 体積保存性型幾何学的発展方程式の解の挙動について
    KOHSAKA YOSHIHITO
    明治非線型数理セミナー one day workshop, Feb. 2018, Japanese, 明治大学中野キャンパス, Domestic conference
    Invited oral presentation

  • Stability and bifurcation diagram of the steady states for the surface diffusion equation
    KOHSAKA YOSHIHITO
    Viscosity solution approach to asymptotic problems in front propagation, dynamical system and related topics, Jul. 2017, English, Research Institute for Mathematical Sciences, International conference
    [Invited]
    Invited oral presentation

  • On the bifurcation diagrams for steady states of the surface diffusion equation
    KOHSAKA YOSHIHITO
    International Conference on PDEs, Geometric Analysis and Functional Inequalities, Mar. 2017, English, University of Sydney, International conference
    [Invited]
    Invited oral presentation

  • 表面拡散方程式の定常問題としての Delaunay 曲面の安定性解析
    KOHSAKA YOSHIHITO
    大阪駅前セミナー, Jan. 2017, Japanese, 龍谷大学大阪梅田キャンパス, Domestic conference
    [Invited]
    Invited oral presentation

  • Bifurcation analysis for steady states of surface diffusion equation
    KOHSAKA YOSHIHITO
    Elucidation of configurational structure in micro behavior and collective pattern formation, Sep. 2016, English, Research Institute for Mathematical Sciences, Kyoto University, International conference
    [Invited]
    Invited oral presentation

  • On steady states for the surface diffusion equation
    KOHSAKA YOSHIHITO
    Workshop on interface motions and free boundary problems: mathematical analysis, numerical analysis, Jul. 2016, English, いするの家 西原脩三記念館, International conference
    [Invited]
    Invited oral presentation

  • 表面拡散方程式の定常曲面の安定性について
    KOHSAKA YOSHIHITO
    東北大学応用数学セミナー, Jun. 2016, Japanese, 東北大学, Domestic conference
    [Invited]
    Public discourse

  • Stability analysis of steady states for surface diffusion equation
    KOHSAKA YOSHIHITO
    Workshop on nonlinear partial differential equations and related topics, May 2016, English, Ishikawa Prefectural Bunkyo Hall, International conference
    [Invited]
    Oral presentation

  • 表面拡散方程式の定常解の安定性解析について
    KOHSAKA YOSHIHITO
    Geometric flows and related problems, Mar. 2016, Japanese, 東京工業大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Stability analysis of the steady states for the surface diffusion equation
    KOHSAKA YOSHIHITO
    松山解析セミナー2016, Feb. 2016, Japanese, 愛媛大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Stability of Delaunay surfaces as the steady states for the surface diffusion equation
    KOHSAKA YOSHIHITO
    Geometric aspects on capillary problems and related topics, Dec. 2015, English, Universidad de Granada, International conference
    [Invited]
    Invited oral presentation

  • 体積保存性をもつ曲面の発展方程式の定常曲面の安定性について
    KOHSAKA YOSHIHITO
    MPM2015, Nov. 2015, Japanese, 宮崎大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Analysis of stability of the CMC surfaces via the surface diffusion equation
    KOHSAKA YOSHIHITO
    Summer School on multiscale and geometric analysis, Jul. 2015, English, Hokkaido University, International conference
    [Invited]
    Invited oral presentation

  • Analysis of stability of the CMC surfaces via the geometric flow with constraints
    KOHSAKA YOSHIHITO
    北陸応用数理研究会2015, Feb. 2015, Japanese, 金沢大学サテライトプラザ, Domestic conference
    [Invited]
    Invited oral presentation

  • 表面拡散方程式と平均曲率一定曲面
    KOHSAKA YOSHIHITO
    パターン形成と界面ダイナミクスの数理, Jan. 2015, Japanese, 京都大学数理解析研究所, Domestic conference
    Keynote oral presentation

  • Stability analysis of stationary surfaces for the geometric flow with constraints
    KOHSAKA YOSHIHITO
    南大阪応用数学セミナー, Jan. 2015, Japanese, 大阪市立大学, Domestic conference
    Public discourse

  • Stability analysis of stationary surfaces for surface diffusion equation
    KOHSAKA YOSHIHITO
    The 32nd Kyushu Symposium on Partial Differential Equations, Jan. 2015, English, Kyushu University Nishijin Plaza, International conference
    [Invited]
    Invited oral presentation

  • Stability analysis of axisymmetric CMC surfaces via surface diffusion equation
    KOHSAKA YOSHIHITO
    東北大学応用数学セミナー, Nov. 2014, Japanese, 東北大学, Domestic conference
    Public discourse

  • Analysis of stability of axisymmetric CMC surfaces via surface diffusion equation
    KOHSAKA YOSHIHITO
    神戸大学解析セミナー, Oct. 2014, Japanese, 神戸大学, Domestic conference
    Public discourse

■ Affiliated Academic Society
  • Mathematical Society of Japan

  • 日本数学会

■ Research Themes
  • ４階非線形放物型偏微分方程式で表される幾何学的発展方程式の解析手法の構築
    高坂 良史, 石井 克幸
    日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2024 - 31 Mar. 2029

  • Canonical mean curvature flow and its application to evolution problems
    利根川 吉廣, 高坂 良史, 石井 克幸, 三浦 達哉, 高棹 圭介, 可香谷 隆, 小野寺 有紹, 水野 将司
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Tokyo Institute of Technology, 01 Apr. 2023 - 31 Mar. 2028

  • 曲率流に対する閾値型近似アルゴリズムとそれを用いた広義解の性質の研究
    石井 克幸, 高坂 良史, 上田 好寛
    日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2023 - 31 Mar. 2026

  • 表面拡散方程式によって時間発展する曲線・曲面の形状と特異性の解析
    高坂 良史, 石井 克幸
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Grant-in-Aid for Scientific Research (C), Kobe University, Apr. 2019 - Mar. 2024, Principal investigator
    Willmore流によって運動が記述される曲面の閾値型近似アルゴリズムの研究を行った。Willmore流の場合は4階熱方程式に対する基本解のTaylor展開をもとに閾値型近似アルゴリズムが導出できることを2020年度に示したので、2021年度はその近似アルゴリズムにおいて基本解の導関数の展開を導くことによって、閾値関数から得られる閾値集合の境界での閾値関数の勾配評価を得た。また、Willmore汎関数に表面積汎関数を加えたエネルギー汎関数の勾配流によって運動が記述される曲面についても閾値型近似アルゴリズムを研究し、4階熱方程式に空間変数に関する2階導関数の項を加えた線形4階放物型偏微分方程式の基本解のTaylor展開をもとにすれば、この場合も閾値型近似アルゴリズムが構成できることを示した。さらにWillmore流の場合と同様にしてその閾値関数から得られる閾値集合の境界での閾値関数の勾配評価を導いた。今回得た閾値型近似アルゴリズムは変分的時間離散近似との関係付けが期待できることに気づいたので、現在は曲げエネルギーに長さ汎関数を加えたエネルギー汎関数の勾配流によって運動が記述される曲線の場合に変分的時間離散近似との関係性をもとにした収束証明を検討している。また、上記の閾値型近似アルゴリズムはHelfrich流(Willmore汎関数に表面積汎関数と体積汎関数を加えたエネルギー汎関数の勾配流)によって運動が記述される曲面への応用も期待できるので、それについても現在検討している。

  • 曲面・曲線からなる曲率流に対する近似アルゴリズムとそれを用いた広義解の性質の研究
    石井 克幸, 高坂 良史, 上田 好寛
    日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2020 - 31 Mar. 2023
    石井は Willmore 流に対する閾値型近似アルゴリズムの研究を行った。4 階熱方程式や 2 階の項をもつ線形 4 階放物型偏微分方程式について、その基本解を Taylor 展開することによって閾値型近似アルゴリズムを構成した。更に 4 階熱方程式の場合には、解の挙動を詳しく調べることにより、その解を使って定義される閾値集合の性質やその境界での解の勾配評価を得た。この証明には半空間に対する定義関数を初期値とする 4 階熱方程式の解のある種の正値性が鍵となっている。2 階の項をもつ線形 4 階放物型偏微分方程式の解を用いて定義される閾値集合の場合にも同様の性質が得られると考えており、研究を進めている。
    高坂はWillmore汎関数に表面積 (曲線の場合は長さ) 汎関数を加えたエネルギー汎関数の勾配流について、閾値型近似アルゴリズムの研究を行った。低階項をもつ線形 4 階放物型偏微分方程式の基本解の Taylor 展開をもとに、上記の勾配流に対する閾値型近似アルゴリズムを構成した。解を用いて定義される閾値集合について、その境界における解の勾配評価について研究を進めている。
    上田はこれまでに研究を進めてきた対称双曲型偏微分方程式系の安定性解析に関する研究に着手し、既知の結果の拡張に成功した。また、一般論の拡張にあたってその最適性に関する問題が浮上したが、京都大学の前川泰則氏と共にその最適性に関する条件の導入にも着手し、ある一定の結果を得ることに成功している。

  • Multifaceted studies on dynamical problems in the calculus of variations using geometric measure theory
    利根川 吉廣, 高坂 良史, 石井 克幸, 三浦 達哉, 高棹 圭介, 可香谷 隆, 小野寺 有紹
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Tokyo Institute of Technology, 01 Apr. 2018 - 31 Mar. 2023
    主な研究成果として以下の２つを挙げる． （１）Salvatore Stuvard（ミラノ大学）と共同で前年度から引き続き研究を行っている余次元１の閉修正可能集合を初期値としたブラッケ流の存在定理について，特にブラッケ流の囲む領域の体積変化について，当初の我々の予想を凌ぐシャープな結果を得ることに成功した．論文はAdvances in Calculus of Variationsに出版受理済である．存在を示したブラッケ流は有界変動関数の枠組みにおける平均曲率流にもなっており，そのためブラッケ流特有の非一意性の問題を解決している．有界変動関数の枠組みの時間大域解の存在定理自体が知られていなかった中，この存在定理は一般化された平均曲率流に対して新しい概念を発見した，とも言える結果である． （２）曲面のブラッケの意味での法線方向速度が，平均曲率と外力項の和で表せる問題を考える．葛西－利根川(2014)による正則性定理により，ほとんどの点において動く曲面はパラボリックの意味で局所的にC^1級グラフになることがわかっていたが，外力項がヘルダー連続ではない場合，界面運動を表す2階放物型方程式の強解になるかどうかは未解決だった．この問題に対し，森龍之介（東工大特別研究員）と富松瑛太（D２）と共同で，もしグラフの時間微分がラドン測度であれば，グラフは期待される２階偏微分方程式の強解になっていることが解明された．興味深いことに，Allen-Cahn方程式から得られる解はこの条件を満たしている．これはブラッケの意味での界面速度の繊細な性質を表すものであり，未知であった現象である．論文はIndiana University Math Journalに出版受理済みである．

  • 石井 克幸
    学術研究助成基金助成金／基盤研究(C), Apr. 2017 - Mar. 2020
    Competitive research funding

  • Pattern dynamics of reaction-diffusion systems and free boundary problems
    Ninomiya Hirokazu, MONOBE Harunori
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Meiji University, 01 Apr. 2014 - 31 Mar. 2018
    To study the spatial patterns of solutions of partial differential equations, such as reaction-diffusion systems, we introduce a reaction-interface system, which consists of the interface equation and an equation in the whole space. This is derived as a singular limit of some reaction-diffusion systems. We studied the multidimensional traveling wave solution and the pulse dynamics of the reaction-interface system. Moreover, for the curvature flow with the anisotropic external force, we study the influence of the anisotropy to the compact traveling wave solutions. We also introduce the layered system to analyze the spatial profiles of solutions in multidimensional space.

  • Variational analysis on dynamic geometric problems
    TONEGAWA Yoshihiro, KIM Lami, WICKRAMASEKERA Neshan
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), 31 May 2013 - 31 Mar. 2018
    We proved the basic existence and regularity theorems for the mean curvature flow considered in the framework of geometric measure theory which is called Brakke's mean curvature flow. As for the existence theorem, when given an arbitrary n-dimensional closed set in an n+1-dimensional Euclidean space, we proved the time global existence of the Brakke's mean curvature flow that evolves from the given initial data. For the analysis of the singular set, we proved the regularity theory around triple junction in the one-dimensional case, and showed the strong stability property of the triple junction within the weak topology of measure.

  • The generalized rotational hypersurfaces and their geomteric evolution problems
    NAGASAWA Takeyuki, TACHIKAWA Atsusi, YAZAKI Shigetoshi, KOHSAKA Yoshihito
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Saitama University, 01 Apr. 2013 - 31 Mar. 2017
    When we deal with variational problems for hypersurfaces, some difficulties may occur. Assuming some symmetry, the problems can be reduced to the problem of curves. Here we obtain three kind of results. First, we reduce the inelastic collision of elastic shell to the problem of centerline, and propose a mathematical model and perform numerical simulation. Secondly, we study the global existence of rotational hypersurface with prescribed mean curvature. The problem can be reduced to the global existence of solutions of an ordinary differential equation which the generating curve satisfies. We resolve the problem completely for all types of hypersurfaces. Lastly we deal with the Moebius energy. We show that the energy can be decomposed into three parts keeping the Moebius invariance. By ude of the decomposition. the variational formulas are derived explicitly. Furthermore we have their estimates on several function spaces.

  • Systematic studies on the profile and behavior of solutions of partial differential equations
    Yanagida Fiji, Polacik Peter, Fila Marek, Shi Jun-Ping, Htoo Khin Phyu Phyu, Hui Kin-Ming, Chern Jann-Long, Chen Chiun-Chuan, Marta Garcia-Huidobro, Lou Yuan
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Tokyo Institute of Technology, 31 May 2012 - 31 Mar. 2017
    We carried out systematic studies on the profile and behavior of solutions of elliptic and parabolic partial differential equations, and obtained the following results among others. For elliptic equations, we showed a Liouville-type theorem for an equation transformed by a scaling method. We also obtained results on the existence and no-existence of singular solutions and the structure of positive solutions for equations on the sphere, and equations with space-dependent exponent of nonlinearities. For parabolic equations, we made clear the structure of solutions of a linear heat equation with inverse-square potential. We also studied the Fujita equation about the boundedness of time-global solutions, existence of singular solutions and their stability, and Fisher's equation about the dynamics of interfaces.

  • 高坂 良史
    学術研究助成基金助成金／基盤研究(C), Apr. 2012 - Mar. 2017, Principal investigator
    Competitive research funding

  • On a dynamics of a solution for geometric evolution equations with a variational structure
    KOHSAKA Yoshihito
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Young Scientists (B), Muroran Institute of Technology, 2008 - 2011
    The motion of three hypersurfaces by mean curvature flow is studied. These hypersurfaces intersect each other. The existence of a local-in-time solution to an initial and boundary value problem for a system of nonlinear parabolic partial differential equations with non-local term is showed. Also, the motion of an axisymmetric surface by surface diffusion equation is studied. In order to obtain the linearized stability of a stationary surface, a corresponding eigenvalue problem is derived and analyzed.

  • Reserch on the stability of solutions of geometric evolution equation using group equivariance
    NAGASAWA Takeyuki, KOIKE Shigeaki, OHTA Masahito, SAKAMOTO Kunio, KOHSAKA Yoshihito, TACHIKAWA Atsushi
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Saitama University, 2005 - 2007
    In this research we investigate the gradient flow with constraints for functionals defined for family of curves and surfaces as geometric evolution equations. The gradient flow, which decreases the value of functionals via deformation, is one of the method for finding critical points. Various shapes in the nature should be stable in some sense. Functionals are the measure of stability, and therefore the limit of gradient flow should be stable in this sense. Nagasawa and Kohsaka consider the Willmore functional for surfaces with prescribed area and enclosed volume (the Helfrich variational problem), and construct the associate gradient flow (the Helfrich flow), and analyze the structure of center manifold near sphere. On the stationary problem for the problem, solutions bifurcating from sphere with more 2, 4, 6 and 8 are constructed by Nagasawa. We reduce the bifurcation equation by use of the group equivariance but not the equivariant branching lemma of the bifurcation theory. Furthermore Nagasawa considers the Helfrich flow for plane curves. There an approximate problem such that the constraints are realized as a singular limit is proposed. It is shown that the uniform estimates for solutions for approximate problem and their convergence. In many case, equations of gradient flow are parabolic type. Koike investigates the maximum principle and comparison results for fully nonlinear parabolic and elliptic equations. Ohta studies the stability of solutions for evolution equation of hyperbolic type. Sakamoto investigates the CR-structure of manifolds. Kohsaka studies the nonlinear stability of stationary solutions for surface diffusion with boundary conditions. Solutions of geometric variational problem are weak solution of a nonlinear equation. Hence it is important to analyze their regularity. Tachikawa studies the regularity theory of minimal critical points for integral functional with discontinuous coefficients.

  • 材料工学にあらわれる非線形問題の数学的な解析
    高坂 良史
    日本学術振興会, 科学研究費助成事業, 若手研究(B), 室蘭工業大学, 2004 - 2006
    平面上の有界領域内で2相を隔てる相境界を表す曲線の運動が幾何学的時間発展方程式の1つである平均曲率流方程式によって記述される数学モデルについて解析を行った。 平均曲率流方程式は、1956年にMullinsによって、粒界の運動方程式として紹介された。また、1979年にはAllenとCahnによって、2元合金の規則相に現れる逆位相境界の運動を記述する界面の運動方程式として平均曲率流方程式が紹介され、そこではGinzburg-Landau型の方程式と平均曲率流方程式との関連性が述べられている。さらに近年では、数理生態学における競争拡散系モデルやBZ反応等の化学反応を記述する反応拡散方程式の特異極限として、平均曲率流方程式が導出されることが知られている。 本年度は、平面上の有界領域内の3重結節点をもつ3相境界運動が平均曲率流方程式によって記述される場合について研究し、この数学モデルの定常曲線(3重結節点をもつ3線分)の非線形安定性の解析を行った。より具体的には、定常曲線の線形安定性の判定基準に用いたエネルギーの第2変分の解析と、幾何学的な考察から得られる曲率に関する方程式に対してエネルギー法的な手法を適用することで、定常曲線の非線形安定性に関する結果を得た。つまり、初期曲線が線形安定(定常曲線の周りでの線形化問題に対応する固有値問題の固有値がすべて負)な定常曲線にエネルギー的視点からみて近い場合に、その初期曲線を出発点とする平均曲率流方程式の解曲線が、この定常曲線に時間発展とともに漸近的に近づいていくことを数学的に解析した。今回適用した手法は、3重結節点をもつ3相境界運動が表面拡散流方程式によって記述される数学モデルの解析にも適用可能であると考えられるので、今後は表面拡散流方程式のモデルに関しても解析を進めていく予定である。

  • Research on special values of the standard L-functions of Siegel modular forms and related fields
    KATSURADA Hidenori, TAKEGAHARA Yugen, HASEGAWA Yuji, KOHSAKA Yoshihito
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Muroran Institute of Technology, 2003 - 2004
    (1)I considered the relation between special values of the standard zeta functions and the congruence a Siegel cusp form of arbitrary degree. In particular, I proposed a conjecture concerning the congruence of the Saito-Kurokawa lift of a cuspidal Hecke eigenform with respect to SL(2,Z) and the special value of its L-function, and proved it under certain condition. Furthermore we gave some numerical evidence for our conjecture. I gave talks on these topics at the workshop held at Hakuba on October 2004, and at the conference held at Karatsu on March 2005. (2)Let M and N be square free positive integers, and φ and χ be Dirichlet characters mod M and N, respectively. For a cuspidal Hecke eigenform f of neben type φ I gave an explicit formula for the special values of the standard zeta function of f twisted by χ. I gave numerical examples for these values and checked some results derived from the Block-Kato conjecture. (3)I proved that the space of Eisenstein series of square free level of quadratic nebentype is spanned by the genus theta series jointly with R. Schulze-Pillot. By using this result, I proved that the space of cusp forms of square free level of quadratic nebentype is spanned by the theta series jointly with S.Boecherer and R.Schulze-Pillot. They gave talks on these topics at the symposium held at Rikkyo University on September 2004. (4)I gave an explicit form of the Koecher-Maass series for the real analytic Eisenstein series jointly with T.Ibukiyama. I gave a talk on this topic at the symposium held at Rikkyo University on September 2004.

  • 材料工学にあらわれる非線形現象の数学的解析
    Competitive research funding

  • Mathematical Analysis of Nonlinear Phenomena
    Competitive research funding

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