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2008 Fiscal Year Annual Research Report

平均場方程式で記述される非線形臨界現象の解析的研究

Research Project

Project/Area Number 20340034
Research InstitutionOsaka University

Principal Investigator

鈴木 貴  Osaka University, 基礎工学研究科, 教授 (40114516)

Co-Investigator(Kenkyū-buntansha) 三沢 正史  熊本大学, 自然科学研究科, 教授 (40242672)
高橋 太  大阪市立大学, 理学部, 准教授 (10374901)
内藤 雄基  神戸大学, 工学部, 准教授 (10231458)
大塚 浩史  宮崎大学, 工学部, 准教授 (20342470)
杉山 由恵  津田塾大学, 学芸学部数学科, 准教授 (60308210)
Keywords解の爆発 / 臨界ソボレフ指数 / タイプIIの爆発レート / コラプスの衝突 / サブコラプスの生成 / 量子化する爆発機構 / Smoluchowski-Poisson方程式 / 双対変分原理
Research Abstract

非平衡熱力学・乱流平均場・自己相互作用流体・ゲージ理論に関わる方程式について,質量保存や自由エネルギー減衰などの変分構造と自己相似性に代表されるスケーリングを用いて解の爆発機構を研究し,特に量子化・爆発レート・サブコラプスの生成・コラプスの衝突に関する統一的なシナリオの糸口を得た.成果の第1は全空間で定義された臨界ソボレフ指数の半線形放物型方程式であり,回転対称性のない場合に爆発の閾値を初めて証明した.次に閾値係数の解の爆発について考察し,有限時間で起こるとすればタイプIIでコラプスの衝突が発生していることを明らかにした.次に重要な成果はkinetic-transport理論から導出される退化放物型方程式の解の爆発機構の量子化に関する先駆的研究であり,変分構造とスケーリング則からまさに空間2次元のSmoluchowski-Poisson方程式の高次元版とみなされること,その一方で質量・自由エネルギーに対峙する2次モーメントの役割が本質的に異なることを明らかにし,新たな解析法を開発してタイプIIの爆発点の有限性などの極めて斬新な結果を得た.この構造はEuler-Poisson方程式・非線形Schr\"odinger方程式にも共通するもので,退化放物型方程式に関するこの研究は半線形放物型方程式に関する前記の成果と合わせて今後の高次元爆発機構解明にひとつの指針を与える仕事と思われる.またAtlantis PressからMean Field Theories and Dual Variationを出版し,これまでの成果を統合して今後の研究の方向を与えた.この中ではとりわけ記憶形状合金方程式について力学安定な自明でない定常解の存在を証明したことが特筆される.

  • Research Products

    (54 results)

All 2009 2008 Other

All Journal Article (25 results) (of which Peer Reviewed: 25 results) Presentation (25 results) Book (2 results) Remarks (1 results) Patent(Industrial Property Rights) (1 results)

  • [Journal Article] Degenerate parabolic equation with critical exponent derived from the kinetic theory, I., generation of theweak solut ion2009

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Adv. Differential Equations 14

      Pages: 433-476

    • Peer Reviewed
  • [Journal Article] Nonlinear and linear conservative finite differenceschemes for regularized long wave equation2009

    • Author(s)
      S. Koide, et al.
    • Journal Title

      Japan J. Indust. Appl. Math. 26

      Pages: 15-40

    • Peer Reviewed
  • [Journal Article] A quasi1inear thermoviscoelastic sys erm for shapememory alloys with temperature dependent specifie heat2009

    • Author(s)
      S. Yoshikawa, et al.
    • Journal Title

      Commun. Pure Appl. Anal. 8

      Pages: 1093-1115

    • Peer Reviewed
  • [Journal Article] Se1f-Simi1arity in chemotaxis systems2008

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Colloquium Mathematics Ill

      Pages: 11-34

    • Peer Reviewed
  • [Journal Article] Variational and seal ing properties of self-interactingfluids and their mass quantizations2008

    • Author(s)
      T. Suzuki
    • Journal Title

      Theoretical and Applied Mechanics Japan 58

      Pages: 207-213

    • Peer Reviewed
  • [Journal Article] Derivation of the equi1ibrium mean field equations ofpoint vortex sys tern and Vortex filament systern2008

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Theoretical and Applied Mechanics Japan 56

      Pages: 285-290

    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of the solution to the non-isothermal phase separation2008

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Nonlinear Analysis 68

      Pages: 1825-1843

    • Peer Reviewed
  • [Journal Article] Local proper ty of the mountain-pass critical point andthe mean field equat ion2008

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Differential and Integral Equations 21

      Pages: 421-432

    • Peer Reviewed
  • [Journal Article] Touchdown and related problems in electrostatic MEMS device eauation2008

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Nonlinear Differential Equations and Applications 15

      Pages: 363-385

    • Peer Reviewed
  • [Journal Article] Weak solutions to the Navier-Stokes-Poisson equation2008

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Adv. Math. Sci. Appl. 18

      Pages: 141-168

    • Peer Reviewed
  • [Journal Article] Semi linear parabol ic equat ion on bounded domain with critical Sobolev exponent2008

    • Author(s)
      T. Suzuki
    • Journal Title

      Indiana Univ. Math. J. 57

      Pages: 3365-3396

    • Peer Reviewed
  • [Journal Article] Capaci ty estimate for the blow-up set of parabolic Equations2008

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Math. Z 259

      Pages: 867-878

    • Peer Reviewed
  • [Journal Article] 移動目標物の追尾制御に関する考察2008

    • Author(s)
      鈴木貴, 他
    • Journal Title

      日本ロボット学会誌 26-8

      Pages: 944-950

    • Peer Reviewed
  • [Journal Article] Regularity condition by mean oscination to a weaksolution of the 2-dimensional Harmonic heat flow intosphere2008

    • Author(s)
      M. Misawa, et al.
    • Journal Title

      Calc. Var. 33

      Pages: 391-415

    • Peer Reviewed
  • [Journal Article] Asymptotic uniqueness of solutions for an elliptic problem with nearly critical growth on symmetric domains2008

    • Author(s)
      F. Takahashi, et al.
    • Journal Title

      Comm. Appl. Nonl. Anal. 15

      Pages: 1-18

    • Peer Reviewed
  • [Journal Article] Asymptotic nondegeneracy of least energy solutions to an elliptic problem with the critical Sobolev exponent2008

    • Author(s)
      F. Takahashi
    • Journal Title

      Advanced Nonlinear Studies 8

      Pages: 783-797

    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of the least energy solutions for abiharmonic problem with nearly critical growth2008

    • Author(s)
      F. Takahashi
    • Journal Title

      Asymptotic Analysis 60

      Pages: 213-226

    • Peer Reviewed
  • [Journal Article] An asymptotic nondegeneracy result for a biharmonicequation with nearly critical growth2008

    • Author(s)
      F. Takahashi
    • Journal Title

      Nonlinear Differential Equations and Applications 15

      Pages: 623-637

    • Peer Reviewed
  • [Journal Article] Sharp conditions for the existence of sign-changingsolutions to equations involving the one-dimensionalp-Laplacian2008

    • Author(s)
      Y. Naito, et al.
    • Journal Title

      Nonlinear Analys is 69

      Pages: 3070-3080

    • Peer Reviewed
  • [Journal Article] Self-simi1ar solutions for a semi1inear heat equationwith critical Sobolev exponent2008

    • Author(s)
      Y. Naito
    • Journal Title

      Indiana Univ. Math. J. 57

      Pages: 1283-1315

    • Peer Reviewed
  • [Journal Article] Local existence and finite time blow-up in the 2-D Keller-Segel system2008

    • Author(s)
      Y. Sugiyaia, et al.
    • Journal Title

      Journal of Evolution Equation 8

      Pages: 353-378

    • Peer Reviewed
  • [Journal Article] Keller-Segel system of parabolic-parabolic type with initialdata in weak $L^(n/2) (R^n)$2008

    • Author(s)
      Y. Sugiyama, et al.
    • Journal Title

      Indiana Univ. Math. J. 57

      Pages: 1467-1500

    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of strong solutions for nonlinearparabolic equations with critical Sobolev exponent2008

    • Author(s)
      M. Ishiwata
    • Journal Title

      Adv. Differential Equations 13

      Pages: 349-366

    • Peer Reviewed
  • [Journal Article] Stationary Fix-Caginalp equation with non-local term

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Nonlinear Analysis (印刷中)

    • Peer Reviewed
  • [Journal Article] Degenerate parabolic equation derived from the kinetictheory

    • Author(s)
      T. Suzuki, et al.
    • Journal Title

      Theoretical and, Applied Mechanics Japan. (印刷中)

    • Peer Reviewed
  • [Presentation] Einstein′s formula and reaction diffusion equations2009

    • Author(s)
      T. Suzuki
    • Organizer
      Mathematical Science and Nonlinear Differential Equations
    • Place of Presentation
      Osaka University
    • Year and Date
      2009-02-14
  • [Presentation] On a nonliearly perturbed elliptic parabolic system in$R^2$2009

    • Author(s)
      黒木場 正城
    • Organizer
      日本数学会九州支部例会
    • Place of Presentation
      福岡大学
    • Year and Date
      2009-01-31
  • [Presentation] Harmonic heat flowと走化性方程式-幾何/解析/物理におけるエネルギーと質量の量子化について2008

    • Author(s)
      鈴木 貴
    • Organizer
      多様体上の微分方程式
    • Place of Presentation
      金沢市
    • Year and Date
      20081202-20081203
  • [Presentation] Geometric diffusion and meand field equations2008

    • Author(s)
      T. Suzuki
    • Organizer
      The 1st Workshop of Shanghai Jiao Tong University and Osaka University on Mathematical Sciences
    • Place of Presentation
      Shanghai Jiao Tong University
    • Year and Date
      2008-12-17
  • [Presentation] Mean field equations provided with mass quantization2008

    • Author(s)
      T. Suzuki
    • Organizer
      PDE approximations in fast reaction-slow diffusion
    • Place of Presentation
      Leiden Univesity
    • Year and Date
      2008-11-12
  • [Presentation] stability of the steady state for multi-dimensionalthermoe1astic systems of shape memory alloys2008

    • Author(s)
      S. Yoshikawa
    • Organizer
      PDE approximations in fast reaction-slow diffusion
    • Place of Presentation
      Leiden Univesity
    • Year and Date
      2008-11-12
  • [Presentation] A variational approach for self-smi1ar solutions to anonlinear heat equation with critical Sobolev exponent2008

    • Author(s)
      Y. Naito
    • Organizer
      偏微分方程式研究集会
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
  • [Presentation] On the Trudinger-Moser ineguality relating to the freeenergy for the mean field of vortices2008

    • Author(s)
      H. Ohtsuka
    • Organizer
      偏微分方程式研究集会
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
  • [Presentation] Onsagerの2次元乱流理論-最近の展開2008

    • Author(s)
      鈴木 貴
    • Organizer
      非線形ダイナミクス研究会
    • Place of Presentation
      大阪大学
    • Year and Date
      2008-10-08
  • [Presentation] stationary Fix-Caginalp equation with non-local term2008

    • Author(s)
      田崎 創平
    • Organizer
      日本数学会総合分科会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-09-26
  • [Presentation] 臨界Sobolev型方程式の爆発解析と漸近的非退化性2008

    • Author(s)
      高橋 太
    • Organizer
      日本数学会総合分科会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-09-26
  • [Presentation] Interfaces for 1-D degenerate Keller-Segel systern2008

    • Author(s)
      杉山 由恵
    • Organizer
      日本数学会総合分科会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-09-26
  • [Presentation] 外部領域におけるstokes半群の重み付きLp-Lq 評価について2008

    • Author(s)
      小林 孝行
    • Organizer
      日本数学会総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2008-09-26
  • [Presentation] Potential well type results for critical semi1inearparabolic eauations with gradient term2008

    • Author(s)
      石渡 通徳
    • Organizer
      日本数学会総合分科会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-09-26
  • [Presentation] ボロノイ格子上の離散変分導関数法2008

    • Author(s)
      降旗 大介
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      東京大学
    • Year and Date
      2008-09-18
  • [Presentation] Self-sdimens2008

    • Author(s)
      Y. Naito
    • Organizer
      Mathematical Analysis on the self-organization and the self-similarity
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-09-17
  • [Presentation] Threshold solutions for a seimi1inear heat eguationwith Sobolev critical exponent2008

    • Author(s)
      Y. Naito
    • Organizer
      Third Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      東北大学
    • Year and Date
      2008-09-12
  • [Presentation] Dimension control of the blowup set for parabolic andelliptic problems2008

    • Author(s)
      T. Suzuki
    • Organizer
      Third Euro-Japanese Workshop on Blowup
    • Place of Presentation
      東北大学
    • Year and Date
      2008-09-11
  • [Presentation] Asymptotic profiles and blowup rates for the $L^2$ critical no nlinear Schr\" odinger equation2008

    • Author(s)
      H. Nawa
    • Organizer
      The Third Euro-Japan Workshop on Blow-up
    • Place of Presentation
      東北大学
    • Year and Date
      2008-09-11
  • [Presentation] Local property of the mountain-pass critical point andthe mean field equation2008

    • Author(s)
      H. Ohtsuka
    • Organizer
      The 5th East China Partial Differential Equations Conference
    • Place of Presentation
      Nanjin University
    • Year and Date
      2008-07-08
  • [Presentation] An analytic approach to the normalized Ricci flow2008

    • Author(s)
      T. Suzuki
    • Organizer
      変分問題とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-25
  • [Presentation] Asymptotic non-degeneracy results for the mean fieldequations on surfaces2008

    • Author(s)
      H. Ohtsuka
    • Organizer
      変分問題とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-25
  • [Presentation] カイネティック理論における退化放物型方程式2008

    • Author(s)
      鈴木 貴
    • Organizer
      第57回理論応用力学連合講演会
    • Place of Presentation
      日本学術会議
    • Year and Date
      2008-06-12
  • [Presentation] Non-homogeneous semi1inear elliptic equations involvingcritical Sobolev exponent2008

    • Author(s)
      Y. Naito
    • Organizer
      International Conference on Nonlinear PDE and Applications
    • Place of Presentation
      Pusan National Univesity
    • Year and Date
      2008-06-11
  • [Presentation] 圧縮性Navier-Stokes-Poisson方程式の緩和時間O極限と退化drift-diffusion方程式2008

    • Author(s)
      小林 孝行
    • Organizer
      第57回理論応用力学講演会
    • Place of Presentation
      日本学術会議
    • Year and Date
      2008-06-10
  • [Book] Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 5, (M. Chipot, ed. ), Nonlineareigenvalue problem with quantization2008

    • Author(s)
      T. Suzuki, et al.
    • Total Pages
      93
    • Publisher
      Elsevier, Amsterdam
  • [Book] Mean Field Theories and Dual Variation2008

    • Author(s)
      T. Suzuki
    • Total Pages
      288
    • Publisher
      Atlantis Press, Amsterdam
  • [Remarks]

    • URL

      http://www.sigmath.es.osaka-u.ac.jp/suzuki/index.html

  • [Patent(Industrial Property Rights)] ホモロジー計算ソフトを用いたガン特定のための画像処理技術2009

    • Inventor(s)
      鈴木 貴, 他2名
    • Industrial Property Rights Holder
      国立大学法人大阪大学
    • Industrial Property Number
      特願2009-015725
    • Filing Date
      2009-01-27

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Published: 2010-06-11   Modified: 2018-05-11  

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