SEARCH

Search Details

TAKAOKA Hideo
Graduate School of Science / Division of Mathematics
Professor

  • Profile

    I am interested in the partial differential equations that describe wave propagation phenomena: nonlinear dispersive and wave equations. In particular, I study local and global in time well-posedness and scattering results with modern theory from functional analysis, harmonic analysis and geometric tools. This research area is still active in which major questions are completely unsolved.

Researcher basic information

■ Research Keyword
  • Partial differntial equation
■ Research Areas
  • Natural sciences / Mathematical analysis

Research activity information

■ Paper
■ MISC
  • Energy transfer model and large periodic boundary value problem for the quintic NLS
    Hideo TAKAOKA
    京都大学数理解析研究所, Nov. 2018, 「数理解析研究所講究緑2093」, 2093(11月号) (11月号), 94 - 103, Japanese
    Meeting report

  • Hideo Takaoka
    2016, COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 41(4) (4), 732 - 747, English
    Technical report

  • フーリエ解析と微分方程式 : 周波数解析の視点から—特集 フーリエ解析の探究 : 現象,理論から普遍へ
    高岡 秀夫
    サイエンス社, Nov. 2015, 数理科学, 53(11) (11), 36 - 42, Japanese

  • 無限次元微積分—特集 無限次元の魅力--諸分野を照らす多彩な舞台装置
    高岡 秀夫
    サイエンス社, Jan. 2010, 数理科学, 48(1) (1), 49 - 54, Japanese

  • Stephen Gustafson, Hideo Takaoka, Tai-Peng Tsai
    Jan. 2009, JOURNAL OF MATHEMATICAL PHYSICS, 50(1) (1), English
    Technical report

  • 非線形分散型波動方程式の大域解析
    高岡 秀夫
    日本数学会, Oct. 2008, 数学 / 日本数学会 編, 60(4) (4), 337 - 351, Japanese

  • 高岡 秀夫
    一般社団法人 日本数学会, 2008, 総合講演・企画特別講演アブストラクト, 2008(Spring-Meeting) (Spring-Meeting), 1 - 12, Japanese

  • テレンス・タオ—ICM関連企画--フィールズ賞業績紹介
    高岡 秀夫
    日本評論社, Feb. 2007, 数学セミナー, 46(2) (2), 42 - 45, Japanese

  • On the infinite dimensional approximation of solution for the KdV equation on the torus(Harmonic Analysis and Nonlinear Partial Differential Equations)
    Takaoka, Hideo
    京都大学数理解析研究所, Jan. 2007, 数理解析研究所講究録, 1529, 110 - 122, Japanese

  • Carlos E. Kenig, Hideo Takaoka
    2006, International Mathematics Research Notices, 2006, English
    Technical report

  • J Colliander, M Keel, G Staffilani, H Takaoka, T Tao
    Jun. 2004, JOURNAL OF FUNCTIONAL ANALYSIS, 211(1) (1), 173 - 218, English
    Technical report

  • Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, Terry Tao
    We obtain global well-posedness, scattering, and global $L^{10}_{t,x}$
    spacetime bounds for energy-class solutions to the quintic defocusing
    Schr"odinger equation in $\R^{1+3}$, which is energy-critical. In particular,
    this establishes global existence of classical solutions. Our work extends the
    results of Bourgain and Grillakis, which handled the radial case. The method is
    similar in spirit to the induction-on-energy strategy of Bourgain, but we
    perform the induction analysis in both frequency space and physical space
    simultaneously, and replace the Morawetz inequality by an interaction variant.
    The principal advantage of the interaction Morawetz estimate is that it is not
    localized to the spatial origin and so is better able to handle nonradial
    solutions. In particular, this interaction estimate, together with an
    almost-conservation argument controlling the movement of $L^2$ mass in
    frequency space, rules out the possibility of energy concentration.
    09 Feb. 2004
    Technical report

  • エネルギー空間より広い空間における非線形シュレディンガー方程式の散乱理論 (非線形波動および分散型方程式に関する研究)
    高岡, 秀夫
    京都大学数理解析研究所, Jan. 2004, 数理解析研究所講究録, 1355, 107 - 116, Japanese

  • 掛谷問題のひろがり--非線形偏微分方程式の視点から—特集 掛谷の問題と実解析
    高岡 秀夫
    日本評論社, Aug. 2002, 数学セミナー, 41(8) (8), 35 - 40, Japanese

  • Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm
    Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, Terence Tao
    We study the long-time behaviour of the focusing cubic NLS on $\R$ in the
    Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper
    bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the
    ground state to polynomial growth at worst; this is a partial analogue of the
    $H^1$ orbital stability result of Weinstein. In the sequel to this paper we
    generalize this result to other nonlinear Schr"odinger equations. Our
    arguments are based on the ``$I$-method'' from our earlier papers, which pushes
    down from the energy norm, as well as an ``upside-down $I$-method'' which
    pushes up from the $L^2$ norm.
    21 Jun. 2002, Discrete Contin. Dynam. Systems, 9, 31 - 54
    Technical report

  • On global well-posedness of some nonlinear dispersive equations for rough data (Harmonic Analysis and Nonlinear P.D.E.)
    Takaoka, Hideo
    Apr. 2001, 1201, 83 - 95, English

  • Time local well-posedness for the KP II equation (Harmonic Analysis and Nonlinear Partial Differential Equations)
    Takakoka, Hideo
    Jun. 1999, 1102, 1 - 8, English

  • TIME LOCAL WELL-POSEDNESS FOR THE ZAKHAROV SYSTEM WITH THE PERIODIC BOUNDARY CONDITION (Harmonic Analysis and Nonlinear Partial Differential Equations)
    Takaoka, Hideo
    Aug. 1998, 1059, 74 - 88, English

■ Lectures, oral presentations, etc.
  • Energy cascades for resonant nonlinear Schrödinger equations
    Hideo TAKAOKA
    九州関数方程式セミナー, Jan. 2019, Japanese, 福岡大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Energy cascades for resonant nonlinear Schrödinger equations
    Hideo TAKAOKA
    Nonlinear Dispersive Equations in Kumamoto, 2019, Jan. 2019, Japanese, 熊本大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Energy transfer model for the resonant 1D quintic NLS
    Hideo TAKAOKA
    Sixth Bielefeld-SNU Joint Workshop in Mathematics, Mar. 2018, English, Seoul National University, International conference
    [Invited]
    Invited oral presentation

  • Energy transfer model and large periodic boundary value problem for the quintic NLS
    Hideo TAKAOKA
    The 15th Linear and Nonlinear Waves, Nov. 2017, English, 大阪大学, International conference
    [Invited]
    Invited oral presentation

  • Energy transfer model and large periodic boundary value problem for the quintic NLS
    Hideo TAKAOKA
    Okayama Workshop on Partial Differential Equations, Oct. 2017, Japanese, 岡山大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Energy transfer model and large periodic boundary value problem for the quintic NLS
    Hideo TAKAOKA
    Nonlinear Wave and Dispersive Equations, Aug. 2017, English, 京都大学, International conference
    [Invited]
    Invited oral presentation

  • Energy transfer model for the resonant nonlinear Schrödinger equations
    Hideo TAKAOKA
    Workshop on linear and nonlinear dispersive equations and related topics, May 2017, English, 京都大学, International conference
    [Invited]
    Invited oral presentation

  • 非線形シュレディンガー方程式に対する共鳴現象と解のダイナミクス
    Hideo TAKAOKA
    談話会, Apr. 2017, Japanese, 神戸大学, Domestic conference
    Oral presentation

  • Energy transfer for the resonant 1D quintic NLS with large periodic boundary condition
    Hideo TAKAOKA
    解析セミナー, Apr. 2017, Japanese, 神戸大学, Domestic conference
    Oral presentation

  • 非線形分散型波動方程式の大域解析
    TAKAOKA Hideo
    2008年度日本数学会年会総合講演, Mar. 2008, Japanese, 日本数学会, 近畿大学, Domestic conference
    [Invited]
    Invited oral presentation

  • Bilinear Strichartz estimates and applications to 2D NLS
    TAKAOKA Hideo
    Nonlinear Wave Equations, Aug. 2007, English, 北海道大学・日本, 北海道大学・日本, International conference
    [Invited]
    Invited oral presentation

  • Bilinear Strichartz estimates and applications to 2D NLS
    TAKAOKA Hideo
    非線形波動および分散型方程式に関する研究, May 2007, Japanese, 京都大学数理解析研究所, 京都大学数理解析研究所, Domestic conference
    [Invited]
    Invited oral presentation

■ Affiliated Academic Society
  • 日本数学会

■ Research Themes
  • 高岡 秀夫
    科学研究費補助金／基盤研究(B), Apr. 2018 - Mar. 2023, Principal investigator
    Competitive research funding

  • 高岡 秀夫
    科学研究費一部基金／基盤研究(B), Apr. 2013 - Mar. 2018, Principal investigator
    Competitive research funding

  • Concentration Phenomena and Structure of Solution for Nonlinear Evolution Equations
    Tsutsumi Yoshio, Okamoto Hisashi, Nakanishi Kenji, Sawano Yoshihiro, Kishimoto Nobu, Takaoka Hideo, Mizoguchi Noriko, Maeda Masaya, Kato Takamori, Miyaji Tomoyuki, Sasaki Youhei
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (S), Kyoto University, 01 Apr. 2011 - 31 Mar. 2016
    For nonlinear wave and dispersive equations, in collaboration with Schlag, Nakanishi succeeded in classifying the global behavior of any solutions starting near an unstable ground state. This is a breakthrough, because there were no results available beofre their papers. Tsutsumi, together with Yoshikawa, proved the existence of two invariant measures for the isothermal Falk model. One is the Gibbs measure and another is an invariant measure proposed by Kuksin. In collaboration with Shoji, Okamoto investigated the Stokes drift of surface gravity waves with surface tension. Okamoto and Shoji proved that the orbits of particles are not closed curves by the method of the complex analysis.

  • Relations between properties of solutions and geometric symmetry of solutions for nonlinear wave and dispersive equations
    TSUTSUMI Yoshio, SHIGEKAWA Ichiro, KOKUBU Hiroshi, NISHIWADA Kimimasa, NAKANISHI Kenji, OKAJI Takashi, OHTA Masahito, TAKAOKA Hideo, TSUGAWA Kotaro, OHTA Masahito, TAKAOKA Hideo, TSUGAWA Kotaro
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Kyoto University, 2007 - 2010
    The research results are as follows.(1) We proved the stability and instability of vortices for the harmonic map heat-flows, the Landau-Lifshitz equations and the Schrodinger maps.(2) We proved the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger with powernonlinearity.(3) We obtained the results about the linear stability and the linear instability of stationary solution bifurcating from the constant stationary solution for the Lugiato-Lefever equation, which is the nonlinear Schrodinger equation with damping and forcing.

  • 内藤 雄基
    科学研究費補助金／基盤研究(C), 2008
    Competitive research funding

  • 足立 匡義
    科学研究費補助金／基盤研究(C), 2008
    Competitive research funding

  • Structure of Solutions and Geometric Symmetry for Nonlinear Evolution Equations
    TSUTSUMI Yoshio, KOKUBU Hiroshi, NAKANISHI Kenji, OHTA Masahito, MIZUMACHI Tetsu, TAKAOKA Hideo
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), 2003 - 2006
    In what follows, the main research results of this project are described. In the academic years of 2003-4, we studied the time local well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. In 1993, Bourgain proved that this Cauchy problem is time locally well-posed in $H^s$, s ≧1/2. Furthermore, in 1996, Bourgain also proved that the solution map fails to belong to $C^3$ in $H^s$, s<1/2, though it is analytic in $H^s$, s≧ 1/2. Takaoka and Tsutsumi made a close investigation into the problem of what is the difference between the cases $H^s$, s ≧1/2 and $H^s$, s<1/2. It was showed that the Cauchy problem is still locally well-posed in $H^s$, s>1/3, but the solution map is not uniformly continuous because of the occurrence of nonlinear oscillation. In the academic year of 2005, Tsutsumi studied the asymptotic behavior of solution for the quadratic nonlinear Schrodinger equation in two space dimensions with Akihiro Shimomura. In two space dimensions, the quadratic nonlinearity is a boader between the short renge case and the long range case and the quadratic nonlinearity has a special interest from a viewpoint of nonlinear scattering theory. It was showed that for a nonlinearity of squared modulous of the unkown, the solution does not approach a free solution, while it is already known that the rest of other quadratic nonlinearities belong to the shourt range interaction. casse. In the academic year of 2006, Tsutsumi studied the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation. When the solution is constructed, one usually impose a condition that the solution belongs to an auxiliary space associated with the Stricahrtz estimate. The unconditional uniqueness means that the solution is unique even though it is not in this kind of auxiliary space. It was proved that the unconditional uniqueness holds for the solution in the critical Sobolev space associated with the scaling invariance of nonlinear SchrOdinger equations. This is a substantial improvement over the results by Furioli and Terraneo.

  • 高岡 秀夫
    科学研究費補助金／若手研究(B), 2006, Principal investigator
    Competitive research funding

  • 非線形分散型方程式の初期値問題の適切性および大域挙動に関する解析的研究
    高岡 秀夫
    日本学術振興会, 科学研究費助成事業, 若手研究(B), 神戸大学, 2003 - 2005
    本年度は、非線形分散型波動現象を記述する非線形シュレディンガー方程式に対して、偏微分方程式論の立場から初期値問題の可解性を研究した。具体的には、プラズマ物理における数学モデルとしても登場する5次の非線形項を持つ非線形シュレディンガー方程式を考えた。トーラス上の初期値問題に対して、最近Bourgainは微分可能指数が1/2よりも小さいSobolev空間で時間大域解の存在定理が成り立つことを証明している。この空間は、不変測度が得られる関数空間として働き、時間大域解の漸近挙動を調べる上でも重要な結果と思われる。一方、全空間の初期値問題に対しては、同一なSobolev空間において時間大域可解性が示されているものの、それよりも小さい微分可能指数での実現の成否は分かっていなかった。全空間とトーラスという設定構造の相違が、解の特異性と非線形相互作用にどのように作用し、初期値問題の適切性にどのような現象を生じるか調べることは興味深い。 フーリエ空間による考察から、非線形相互作用によって周波数の停滞する項が現れることが分かり、先行の解の関数空間はその項の処理に依存していた。トーラス上の初期値問題では、解の関数空間に解自身に依存する構造を取り入れ、問題となる停滞項を取り除くことが行われたが、全空間の場合はそれが連続的であり、トーラス上の問題と同様な処理は困難と思われる。今回は、平滑効果を導く新しい3線形評価式を示すことからその様な周波数の停滞は瞬時に起こることを示し、時間大域解の存在証明に対して微分可能回数がこれまでよりも小さい関数空間で評価できる証明を与えた。如何に時間局所解が大域的に延長されるかは未解決として残っているが、他の方程式への応用は十分期待される。この成果は論文として現在取りまとめている。

  • GEOMETRY AND ANALYSIS FOR WAVE FIELDS
    OZAWA Tohru, NAKAMURA Gen, TSUTSUMI Yoshio, HAYASHI Nakao, NAKANISHI Kenji, TAKAOKA Hideo
    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Hokkaido University, 2001 - 2003
    In this research project, various space-time behavior of solutions to nonlinear dispersive equations, such as nonlinear Schrodinger equations (NLS) and KdV type equations, nonlinear hyperbolic equations, such as nonlinear wave and Klein-Gordon equations, and coupled systems of those equations, such as nonlinear field equations. The main results are the following : (1)Asymptotic completeness in the energy space H^1(R^3) for NLS with repulsive case has been proved. (2)A unified treatment for small data scattering for nonlinear field equations has been given in terms of critical and subcritical setting. (3)Existence and uniqueness of self-similar solutions for nonlinear wave equations have been proved in the framework of weak Lebesgue spaces.

  • 非線形分散型方程式の初期値問題の可解性とその解の性質について
    高岡 秀夫
    日本学術振興会, 科学研究費助成事業, 若手研究(B), 2001 - 2002
    偏微分方程式論の関数解析的研究の立場から、特にKdV方程式や非線形シュレディンガー方程式などの非線形分散型波動方程式の数学解析を行った。分散型方程式において、解の特異性がどのように非線形相互作用するかを調べることは重要な研究である。その研究成果の一端として最近、非線形分散型波動方程式について、十分な滑らかさを伴わない初期値でも解の存在定理やその解の性質を調べることが行われ、その研究の概観作りがなされつつある。その中で本年度の研究は、周期境界値条件における修正KdV方程式と空間3次元の非線形シュレディンガー方程式に対して、方程式の初期値問題の適切性が数学的にどのように実現され得るかに着目し非線形問題の理解を目指した。 周期境界値条件における修正KdV方程式に対して、初期値に対応する特解の周りで解の摂動問題を考え、これまでに捉えきれなかった解の存在定理を証明した。この証明法は方程式の幾何学的対称性を用いて方程式の性質を調べることとも密接な関係があり、他の方程式の同様な問題に対する適用範囲の拡大が期待される。空間3次元の非線形シュレディンガー方程式に対しては、既知とされる時間局所解の研究を出発点とし、解の延長問題とその大域的挙動の問題が検討された。解の接続問題を扱うところではこれまでに確立したエネルギー輸送という方法論の適用を行ったが、更にここでは新しいタイプのエネルギー減衰評価式を提出し、この2つの議論の融合から波の散乱現象の実現を成功させ、時刻無限大で解は自由解に近づくことを証明した。これらの研究成果については論文として現在執筆中である。 最後に、今年度の研究課題を遂行するにあたって、昨年度と同様に国内外への研究調査費、図書費、文具費ならび謝金として研究補助金が有効に使用されたことを報告する。

TOP