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

2012 Fiscal Year Annual Research Report

• Project/Area Number: 20340034
• Research Institution: Osaka University
• Principal Investigator: 鈴木 貴 大阪大学, 基礎工学研究科, 教授 (40114516)
• Co-Investigator(Kenkyū-buntansha): 石渡 通徳 福島大学, 共生システム理工学類, 准教授 (30350458) ; 佐藤 友彦 日本大学, 生産工学部, 助教 (50397676)
• Project Period (FY): 2008-04-08 – 2013-03-31
• Keywords: 記憶形状物質 / 競合的走化性 / Lotka-Volterra系 / 点渦乱流平均場 / Gel’fand 方程式 / 退化放物型方程式 / Smoluchowski-Poisson方程式 / 量子化爆発機構

Research Abstract

本課題は, 統計力学や熱力学を基盤として自然科学の様々な分野で導出される非線形, 非平衡, 多種の数理モデルに対し, スケーリングや変分構造を用いて互いに関連付けて解析して, 爆発や界面など共通の現象を数学的に解明することを目指したものである. 本年度は最終年度であり, これまでに取り上げてきたいくつかのテーマを完成させるとともに, 新たな問題や解析方法も提出した. 特筆すべきものの第1は, 場と粒子に関わる双対的な変分構造のもとで保障される定常解の安定性についてである. 本年度は特に熱弾性記憶形状物質モデルでもこの原理が成り立つこと, さらに解析的非線形性の下では縮約汎関数の極小は無限小安定であることを証明した. 第2は多成分系の相互作用が及ぼす解の挙動である. すなわち, 腫瘍微小環境で現出する競合的走化性の下で1種のみがデルタ関数を形成すること（質量分離）, 生態系モデルで解が時間とともに空間的一様な状態に遷移すること, 渦度が連続的に分布する点渦乱流平均場における逆温度の臨界値が存在することを, 厳密に解明した. 第3はスケーリングと爆発機構の量子化に関わるものである. 特に, 空間2次元のGel’fand方程式の多点爆発解において解の線形化非退化性がその爆発点を規定するハミルトニアンの臨界点の非退化性から導出されること, 高次元の退化放物型方程式において臨界指数の場合にタイプIIの爆発点が有限個であること, 空間2次元のSmoluchowski-Poisson方程式において, Dirichlet条件下では境界上の爆発点が存在しないことを証明した

Current Status of Research Progress

Reason

24年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

24年度が最終年度であるため、記入しない。

Research Products

• [Journal Article] Asymptotic non-degeneracy of multiple blowup solution to the Liouville-Gel’fand problem with non-constant coefficient (2013)
  • Author(s): H. Ohtsuka, T. Sato, and T. Suzuki
  • Journal Title: J. Math. Anal. Appl. Volume: 398 Pages: 692-706
  • DOI: 10.1016
  • Peer Reviewed
• [Journal Article] Non-local reaction-diffusion system involving reaction radius II, rate of convergence (2013)
  • Author(s): N. Kavallaris and T. Suzuki
  • Journal Title: IMA J. Appl. Math Volume: 0
  • DOI: 10.1093
  • Peer Reviewed
• [Journal Article] Global-in-time behavior of the solution to a Gierer-Meihardt system (2013)
  • Author(s): G. Karali, T. Suzuki, and Y. Yamada
  • Journal Title: Discrete and Continous Dynamical Systems Volume: A
  • DOI: 103934
  • Peer Reviewed
• [Journal Article] A blowup criterion for compressible isentropic Euler equation (2013)
  • Author(s): T. Suzuki
  • Journal Title: J. Fluid Mech Volume: 0
  • DOI: 10.1007
  • Peer Reviewed
• [Journal Article] Duality and best constant for a Trudinger-Moser inequality involving probability measures (2013)
  • Author(s): T. Ricciardi and T. Suzuki
  • Journal Title: Euro J. Appl. Anal Volume: 0
  • Peer Reviewed
• [Journal Article] Exclusion of boundary blowup for $2D$ chemotaxis system provided with Dirichlet boundary condition for the Poisson part (2013)
  • Author(s): T. Suzuki
  • Journal Title: J. Math. Pure Appl Volume: 0
  • Peer Reviewed
• [Journal Article] Asymptotic non-degeneracy of multiple blowup solutions to the Liouville-Gel'fand problem with an inhomogeneous coefficient (2013)
  • Author(s): Hiroshi Ohtsuka, Tomohiko Sato, Takashi Suzuki
  • Journal Title: Journal of Mathematical Analysis and Volume: 398, Issue 2 Pages: 692,-706
  • DOI: 10.1016/j.jmaa.2012.09.028
  • Peer Reviewed
• [Journal Article] Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations (2013)
  • Author(s): Yuki Kaneko, Kazuhiro Oeda And Yoshio Yamada
  • Journal Title: Funkcialaj Ekvacioj Volume: 0
  • Peer Reviewed
• [Journal Article] On the initial value problem of the semilinear beam equation with weak damping I: smoothing effect (2013)
  • Author(s): Hiroshi Takeda, Shuji Yoshikawa
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 401 Pages: 244-258
  • DOI: 10.1016/j.jmaa.2012.12.015
  • Peer Reviewed
• [Journal Article] Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space (2013)
  • Author(s): Yuki Naito and Takasi Senba
  • Journal Title: Commun. Pure Appl. Anal. Volume: 12
  • DOI: 10.3934/cpaa.2013.12.
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of eigenvalues of the Laplacian with the mixed boundary condition and its application (2013)
  • Author(s): Shuichi Jimbo and Kazuhiro Kurata
  • Journal Title: 数理解析研究所講究録 Volume: 0
• [Journal Article] Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species (2012)
  • Author(s): E.E. Espejo, A. Stevens, and T. Suzuki
  • Journal Title: Differential and Integral Equations Volume: 25 Pages: 251-288
  • Peer Reviewed
• [Journal Article] Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys (2012)
  • Author(s): T. Suzuki and S. Yoshikawa
  • Journal Title: Discrete and Continuous Dynamical Systems S 5-1 Volume: Special Issue Pages: 209-217
  • DOI: 10.3934
  • Peer Reviewed
• [Journal Article] Non-local reaction-diffusion system involved by reaction radius I (2012)
  • Author(s): N. Kavallaris and T. Suzuki
  • Journal Title: IMA J. Appl. Math. Volume: 0 Pages: 1-19
  • DOI: 10.1093
  • Peer Reviewed
• [Journal Article] Stationary solutions to a strain-gradient type thermoviscoelastic system (2012)
  • Author(s): I . Paw\l ow, T. Suzuki, and S. Tasaki
  • Journal Title: Differential and Integral Equations Volume: 25 Pages: 289-340
  • Peer Reviewed
• [Journal Article] Degenerate parabolic equation with critical exponent derived from the kinetic theory, III., $\varepsilon$-regularity (2012)
  • Author(s): T. Suzuki and R. Takahashi
  • Journal Title: Differential and Integral Equations Volume: 25 Pages: 223-250
  • Peer Reviewed
• [Journal Article] Transient and asymptotic dynamics of a prey-predator system with diffusion (2012)
  • Author(s): E. Latos, T. Suzuki, and Y. Yamada
  • Journal Title: Math. Meth. Appl. Sci Volume: 35 Pages: 1101-1109
  • DOI: 10.1002
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law with partially linearly degenerate flux (2012)
  • Author(s): A. Matsumura and N. Yoshida
  • Journal Title: SIAM J. Math. Anal. Volume: Vol. 44, No. 4 Pages: 2526-2544
  • DOI: 10.1137/110839448
  • Peer Reviewed
• [Journal Article] 交差拡散を伴う非線形拡散方程式系 --数理生態学に現れる反応拡散方程式系-- (2012)
  • Author(s): 山田 義雄
  • Journal Title: 数学 Volume: 64 Pages: 384-406
  • Peer Reviewed
• [Journal Article] On limit systems for some population models with cross-diffusion (2012)
  • Author(s): Kousuke Kuto and Yoshio Yamada
  • Journal Title: Discrete and Continuous Dynamical Systems B Volume: 17 Pages: 2745-2769
  • DOI: 10.3934/dcdsb.2012.17.2745
  • Peer Reviewed
• [Journal Article] On the initial value problem of the semilinear beam equation with weak damping II: asymptotic profiles (2012)
  • Author(s): Hiroshi Takeda, Shuji Yoshikawa
  • Journal Title: Journal of Differential Equations Volume: 253 Pages: 3061-3080
  • DOI: 10.1016/j.jde.2012.07.014
  • Peer Reviewed
• [Journal Article] The role of forward self-similar solutions in the Cauchy problem for semilinear heat equations (2012)
  • Author(s): Yuki Naito
  • Journal Title: J. Differential Equations Volume: 253 Pages: 3029-3060
  • DOI: dx.doi.org/10.1016/j.jde.2012.08.013
  • Peer Reviewed
• [Journal Article] Remarks on singular Sturm comparison theorems (2012)
  • Author(s): Yuki Naito
  • Journal Title: Mem. Differential Equations Math. Phys. Volume: 57 Pages: 109-122
  • Peer Reviewed
• [Journal Article] Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains (2012)
  • Author(s): Yuki Naito and Takasi Senba
  • Journal Title: Discrete Contin. Dyn. Syst. Volume: 32 Pages: 3691-3713
  • DOI: 10.3934/dcds.2012.32.xx
  • Peer Reviewed
• [Journal Article] Classification of second order linear differential equations and an application to singular elliptic eigenvalue problems (2012)
  • Author(s): Yuki Naito
  • Journal Title: Bull. Lond. Math. Soc. Volume: 44 Pages: 545-562
  • DOI: 10.1112/blms/bdr117
  • Peer Reviewed
• [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent (2012)
  • Author(s): Yuki Naito and Tokushi Sato
  • Journal Title: Ann. Mat. Pura Appl. (4) Volume: 191 Pages: 25-51
  • DOI: 10.1007/s10231-010-0173-y
  • Peer Reviewed
• [Journal Article] Rectifiable oscillations of radially symmetric solutions of p-Laplace differential equations (2012)
  • Author(s): Yuki Naito, Mervan Pasic, and Hiroyuki Usami
  • Journal Title: Differ. Equ. Appl. Volume: 4 Pages: 11-25
  • DOI: dx.doi.org/10.7153/dea-04-03
  • Peer Reviewed
• [Presentation] Global behavior of solutions to semilinear heat equations (2013)
  • Author(s): 内藤雄基
  • Organizer: International Conference "Quantization blow-up and concentration in mathematical physics view point"
  • Place of Presentation: 大阪大学
  • Year and Date: 20130328-20130328
  • Invited
• [Presentation] 非線形熱方程式の解の漸近挙動と自己相似解 (2013)
  • Author(s): 内藤雄基
  • Organizer: 神戸解析セミナー
  • Place of Presentation: 神戸大学
  • Year and Date: 20130205-20130205
  • Invited
• [Presentation] Lotka-Vorterra系の時間大域挙動 (2013)
  • Author(s): T. Suzuki
  • Organizer: 共同研究「非線型現象の数理モデルに現れる近平衡解の時間的漸近挙動に関する研究」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 20130129-20130129
• [Presentation] Design to construct weakly nonlinear schemes in structure preserving context (2013)
  • Author(s): D.Furihata
  • Organizer: 2013 Tokyo Workshop on Structure-Preserving Methods
  • Place of Presentation: Tokyo, Japan
  • Year and Date: 20130108-20130108
  • Invited
• [Presentation] Spreading and vanishing dichotomy for some free boundary problems in ecology (2012)
  • Author(s): Yoshio Yamada
  • Organizer: Swiss-Japanese Seminar
  • Place of Presentation: チューリッヒ大学
  • Year and Date: 20121218-20121218
  • Invited
• [Presentation] Separation structure of positive radial solutions for semilinear elliptic equations (2012)
  • Author(s): 内藤雄基
  • Organizer: Swiss-Japanese Seminar
  • Place of Presentation: スイス・チューリッヒ大学
  • Year and Date: 20121218-20121218
  • Invited
• [Presentation] On logistic equations with diffusion and nonlocal effects (2012)
  • Author(s): Yoshio Yamada
  • Organizer: Workshop on Nonlocal Problems—in the framework of the EU programme FIRST
  • Place of Presentation: チューリッヒ大学
  • Year and Date: 20121214-20121214
  • Invited
• [Presentation] Spreading and vanishing for free boundary problems in ecology (2012)
  • Author(s): Yoshio Yamada
  • Organizer: 5th Polish-Japanese Days on Nonlinear Analysis in Interdisciplinary Sciences
  • Place of Presentation: 京都、関西セミナーハウス
  • Year and Date: 20121109-20121109
  • Invited
• [Presentation] Asymptotic behavior of eigenvalues of the Laplacian with the mixed boundary condition and its application (2012)
  • Author(s): Kazuhiro Kurata
  • Organizer: RIMS研究集会「偏微分方程式の解の幾何」
  • Place of Presentation: 数理解析研究所
  • Year and Date: 20121109-20121109
  • Invited
• [Presentation] Smoluchowski-Poisson equation in statistical physics and cell biology - mathematics for blowup with quantization (2012)
  • Author(s): T. Suzuki
  • Organizer: 5th Polish-Japanese Days on Nonlinear Analysis in Interdisciplinary Sciences, Modelling, Theory and Simulations
  • Place of Presentation: Kansai Seminar House, Kyoto
  • Year and Date: 20121106-20121106
• [Presentation] Asymptotic profiles of solutions for the isothermal Falk-Konopka system of shape memory alloys with weak damping (2012)
  • Author(s): Shuji Yoshikawa
  • Organizer: Linear and Nonlinear Waves, No.10
  • Place of Presentation: Piazza Ohmi (大津市)
  • Year and Date: 20121101-20121101
  • Invited
• [Presentation] 非線形熱方程式の解の漸近挙動と自己相似解 (2012)
  • Author(s): 内藤雄基
  • Organizer: 熊本大学数学談話会
  • Place of Presentation: 熊本大学
  • Year and Date: 20121031-20121031
  • Invited
• [Presentation] Some results by energy methods on large-time behavior of viscous gas (2012)
  • Author(s): Akitaka Matumura
  • Organizer: 2012 International Conference on Nonlinear Analysis： Evolutionary
  • Place of Presentation: Institute of Mathematics,Academia Sinica, Taipei
  • Year and Date: 20121029-20121102
  • Invited
• [Presentation] Higher order expansion of solutions for the isothermal Falk-Konopka system of shape memory alloys (2012)
  • Author(s): 吉川周二
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 20121013-20121013
  • Invited
• [Presentation] Global attractivity of self-similar solutions for semilinear heat equations (2012)
  • Author(s): 内藤雄基
  • Organizer: 研究集会「広島微分方程式研究会」
  • Place of Presentation: 広島大学
  • Year and Date: 20121006-20121006
  • Invited
• [Presentation] 形状記憶合金の等温Falk-Konopkaモデルの解の高次漸近形 (2012)
  • Author(s): 吉川周二
  • Organizer: 広島微分方程式研究会
  • Place of Presentation: 広島大学
  • Year and Date: 20121005-20121005
  • Invited
• [Presentation] Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part (2012)
  • Author(s): T. Suzuki
  • Organizer: 日本数学会総合分科会
  • Place of Presentation: 九州大学伊都キャンパス
  • Year and Date: 20120920-20120920
• [Presentation] A new technique to design numerical schemes with weak nonlinearity based on discrete variational derivative method (2012)
  • Author(s): D. Furihata
  • Organizer: International conference of numerical analysis and applied mathematcs
  • Place of Presentation: Kos, Greece
  • Year and Date: 20120920-20120920
• [Presentation] Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part (2012)
  • Author(s): T. Suzuki
  • Organizer: PDE seminar
  • Place of Presentation: Max Planck Institute (Mathematics), Leipzig
  • Year and Date: 20120917-20120917
• [Presentation] Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part (2012)
  • Author(s): T. Suzuki
  • Organizer: 5th Euro-Japan Conference on Blowup
  • Place of Presentation: CIRM, Marseille
  • Year and Date: 20120913-20120913
• [Presentation] Global attractivity properties of stationary solutions for semilinear heat equations (2012)
  • Author(s): 内藤雄基
  • Organizer: 5th Euro-Japanese Workshop on Blow-up
  • Place of Presentation: フランス、マルセイユ
  • Year and Date: 20120910-20120910
  • Invited
• [Presentation] 問題の非線形性を弱める離散変分スキームの構成 (2012)
  • Author(s): 降籏 大介
  • Organizer: 日本応用数理学会2012年度年会
  • Place of Presentation: 稚内, 日本
  • Year and Date: 20120829-20120829
• [Presentation] Weakly nonlinear structure preserving schemes for strongly nonlinear partial differential equations (2012)
  • Author(s): D. Furihata and T. Matsuo
  • Organizer: International Congress on Computational and Applied Mathematics
  • Place of Presentation: Gent, Belgium
  • Year and Date: 20120710-20120710
• [Presentation] A note on the stabilization of multistep linearly-implicit schemes for dissipative systems (2012)
  • Author(s): T. Matsuo and D. Furihata
  • Organizer: International Congress on Computational and Applied Mathematics
  • Place of Presentation: Gent, Belgium
  • Year and Date: 20120710-20120710
• [Presentation] Asymptotic profiles of solutions for the isothermal Falk-Konopka system with weak damping (2012)
  • Author(s): Shuji Yoshikawa
  • Organizer: The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Orland, Florida, USA
  • Year and Date: 20120703-20120703
  • Invited
• [Presentation] 形状記憶合金の等温Falk-Konopkaモデルの解の三次の漸近形 (2012)
  • Author(s): 吉川周二
  • Organizer: 京都駅前セミナー
  • Place of Presentation: キャンパスプラザ京都
  • Year and Date: 20120615-20120615
  • Invited
• [Presentation] Trudinger-Moser inequality for a point vortex mean field limit with multi-intensities (2012)
  • Author(s): T. Suzuki
  • Organizer: RIMS研究集会 変分問題の展開-確率論と交錯する変分問題
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 20120611-20120611
• [Presentation] Separation structure of positive radial solutions for semilinear elliptic equations (2012)
  • Author(s): 内藤雄基
  • Organizer: 平成24年度第2回「南大阪応用数学セミナー」
  • Place of Presentation: 大阪府立大学
  • Year and Date: 20120526-20120526
  • Invited
• [Presentation] 2D Smoluchowski-Poisson equation - method of the weak scaling limit (2012)
  • Author(s): T. Suzuki
  • Organizer: Seminar in Laboratories of Mathematics, Univ. Paul Sabatier
  • Place of Presentation: Toulouse
  • Year and Date: 20120502-20120502