Studies on well-posedness for parabolic equations with nonlinear constraints

• Project/Area Number: 23540206
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hiroshima University
• Principal Investigator: MATSUMOTO TOSHITAKA 広島大学, 理学(系)研究科(研究院), 助教 (20229561)
• Co-Investigator(Kenkyū-buntansha): KOBAYASHI Yoshikazu 中央大学, 理工学部, 教授 (80092691) ; WATANABE Hiroshi サレジオ工業高等専門学校, 助教 (30609912)
• Co-Investigator(Renkei-kenkyūsha): TANAKA Naoki 静岡大学, 大学院・理学研究科, 教授 (00207119)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2011: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 適切性 / 放物型方程式 / リプシッツ作用素半群 / 非線形境界条件

Research Abstract

We studied well-posedness for parabolic equations and generation theorems of semigroups of Lipschitz operators. A nonlinear perturbation theory for analytic semigroups was extended to the case of analytic semigroups of growth order alpha. The obtained results were applied to showing the well-posedness of 2-D drift-diffusion equations with no-flux boundary conditions. Existence and uniqueness results for weak solutions to strongly degenerate parabolic equations and a phase filed model of grain boundary motion were also derived. Futhermore, we provided an approximation theorem for semigroups of Lipschitz operators and its application to 2-D Navier-Stokes equations as well as a characterization theorem for Lipschitz evolution operators in Banach spaces.

Research Products

• [Journal Article] Solvability of boundary value problems for strongly degenerate parabolic equaions with discontinuous coefficients (2014)
  • Author(s): Hiroshi Watanabe
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. S Volume: 7 Issue: 1 Pages: 177-189
  • DOI: 10.3934/dcdss.2014.7.177
  • Peer Reviewed
• [Journal Article] Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion (2014)
  • Author(s): K. Shirakawa and H. Watanabe
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. S Volume: 7 Issue: 1 Pages: 139-159
  • DOI: 10.3934/dcdss.2014.7.139
  • Peer Reviewed
• [Journal Article] Entropy solutions to strongly degenerate parabolic equations with zero-flux boundary conditions (2013)
  • Author(s): H. Watanabe
  • Journal Title: Adv. Math. Sci. Appl Volume: 23 Pages: 209-234
  • Peer Reviewed
• [Journal Article] Qualitative properties of a one-dimensional phase-field system associated with grain boundary, GAKUTO Internat (2013)
  • Author(s): H. Watanabe and K. Shirakawa
  • Journal Title: Ser. Math. Sci. Appl Volume: 36 Pages: 301-328
  • Peer Reviewed
• [Journal Article] Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients (2013)
  • Author(s): H. Watanabe
  • Journal Title: Dynamical Sytems, Differential Equations, DCDS Supplement 2013, Proceedings of the 9th AIMS Conference, Orland Florida, USA Pages: 781-790
  • URL: http://www.aimsciences.org/journals/pdfs.jsp?paperID=9260&mode=full
  • Peer Reviewed
• [Journal Article] Approximation of semigroups of Lipschitz operators (2013)
  • Author(s): Yoshikazu Kobayashi, Naoki Tanaka
  • Journal Title: J. Math. Anal. Appl Volume: 397 Issue: 2 Pages: 529-536
  • DOI: 10.1016/j.jmaa.2012.07.069
  • Peer Reviewed
• [Journal Article] Nonlinear perturbations of a class of holomorphic semigroups of growth order α by comparison theorems for Volterra equations (2013)
  • Author(s): Toshitaka Matsumoto, Naoki Tanaka
  • Journal Title: Nonlinear Anal Volume: 84 Pages: 146-175
  • DOI: 10.1016/j.na.2013.02.016
  • Peer Reviewed
• [Journal Article] Solvability for one-dimensional phase field system associated with grain boundary motion (2013)
  • Author(s): K. Shirakawa, H. Watanabe and N. Yamazaki
  • Journal Title: Math. Ann Volume: 356 Issue: 1 Pages: 301-330
  • DOI: 10.1007/s00208-012-0849-2
  • Peer Reviewed
• [Journal Article] Entropy solutions to strongly degenerate parabolic equations with zero-flux boundary conditions (2013)
  • Author(s): Hiroshi Watanabe
  • Journal Title: Advances in Mathematical Sciences and Applications Volume: 23
  • Peer Reviewed
• [Journal Article] Qualitative properties of a one-dimensional phase-field system associated with grain boundary (2013)
  • Author(s): Hiroshi Watanabe and Ken Shirakawa
  • Journal Title: GAKUTO Internat. Ser. Math. Sci. Appl. Volume: 36
  • Peer Reviewed
• [Journal Article] Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients (2013)
  • Author(s): Hiroshi Watanabe
  • Journal Title: Dynamical Systems, Differential Equations and Applications Volume: DCDS Supplement 2013
  • Peer Reviewed
• [Journal Article] 不連続な係数を持つ強退化放物型方程式に対する初期値境界値問題 (2012)
  • Author(s): 渡邉紘
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1810 Pages: 153-168
  • URL: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1810-08.pdf
• [Journal Article] 結晶粒界現象に関連する 1 次元フェーズ・フィールドモデル (2012)
  • Author(s): 白川健, 渡邉紘, 山崎教昭
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1779 Pages: 27-51
  • URL: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1779-04.pdf
• [Journal Article] 結晶粒界現象に関連する１次元フェーズ・フィールドモデル (2012)
  • Author(s): 白川健
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1779 Pages: 27-51
• [Journal Article] Solvability for one-dimensional phase field system associated with grain boundary motion (2011)
  • Author(s): K. Shirakawa, H. Watanabe and N. Yamazaki
  • Journal Title: Technical Reports of Mathematical Sciences, Chiba University Volume: 27 Pages: 1-29
  • URL: http://www.math.s.chiba-u.ac.jp/report/files/11001.pdf
• [Journal Article] 不連続な係数を持つ強退化放物型方程式に対する初期値問題について (2011)
  • Author(s): 渡邉 紘
  • Journal Title: サレジオ工業高等専門学校紀要 Volume: 38 Pages: 13-20
  • NAID: 110008803256
  • URL: http://ci.nii.ac.jp/naid/110008803256
• [Journal Article] Solvability for one-dimensional phase field system associated with grain boundary motion, (2011)
  • Author(s): K.Shirakawa, H.Watanabe and N.Yamazaki,
  • Journal Title: Technical Reports of Mathematical Siences Chiba University Volume: 27
• [Journal Article] 不連続な係数を持つ強退化放物型方程式に対する初期値問題について (2011)
  • Author(s): 渡邉 紘
  • Journal Title: サレジオ工業高等専門学校研究紀要 Volume: 38
  • NAID: 110008803256
• [Presentation] 空間変数に依存する拡散項を持つ強退化放物型方程式 (2014)
  • Author(s): 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 学習院大学(東京都)
  • Year and Date: 2014-03-18
• [Presentation] 結晶粒界現象に対する凝固による界面拡散を考慮した数学モデル (2014)
  • Author(s): 白川健, 渡邉紘, 山崎教昭
  • Organizer: 日本数学会
  • Place of Presentation: 学習院大学(東京都)
  • Year and Date: 2014-03-18
• [Presentation] 結晶粒界現象に対する凝固による界面拡散を考慮した数学モデル (2014)
  • Author(s): 白川健, 渡邉紘, 山崎教昭
  • Organizer: 日本数学会
  • Place of Presentation: 学習院大学
• [Presentation] 空間変数に依存する拡散項を持つ強退化放物型方程式 (2014)
  • Author(s): 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 学習院大学
• [Presentation] 退化性を伴う 1 次元Kobayashi-Waren-Carter モデルに対する定性的性質 (2013)
  • Author(s): 渡邉紘, 白川健
  • Organizer: 第39回発展方程式研究会
  • Place of Presentation: 日本女子大学(東京都)
  • Year and Date: 2013-12-23
• [Presentation] 結晶粒界現象を記述する退化性を伴う 1 次元数学モデルに対する定性的性質 (2013)
  • Author(s): 渡邉紘, 白川健
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学(愛媛県)
  • Year and Date: 2013-09-25
• [Presentation] 結晶粒界現象の放物型フェーズ・フィールドモデルにおける平滑化効果とエネルギー消散性 (2013)
  • Author(s): 白川健, 渡邉紘, S. Moll
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学(愛媛県)
  • Year and Date: 2013-09-25
• [Presentation] バナッハ空間における非線形発展作用素と常微分方程式 (2013)
  • Author(s): 小林良和
  • Organizer: 日本数学会企画特別講演
  • Place of Presentation: 愛媛大学(愛媛県)
  • Year and Date: 2013-09-24
• [Presentation] Energy-dissipations in multidimensional Kobayashi-Waren-Carter models of grain boundaries (2013)
  • Author(s): H. Watanabe
  • Organizer: Equadiff 13
  • Place of Presentation: Charles University in Prague (チェコ)
• [Presentation] 強退化放物型方程式に対する動力学的接近 (2013)
  • Author(s): 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学(京都府)
• [Presentation] Banach空間上のLipschitz発展作用素 (2013)
  • Author(s): 冨澤佑季乃, 小林良和, 田中直樹
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学(京都府)
• [Presentation] 強退化放物型方程式の初期値境界値問題に対する解の一意存在性 (2013)
  • Author(s): 渡邉紘
  • Organizer: 第5回東北楕円放物型微分方程式研究集会
  • Place of Presentation: 東北大学(宮城県)
• [Presentation] Energy-dissipations in multidimensional Kobayashi-Warren-Carter models of grain boundaries (2013)
  • Author(s): Hiroshi Watanabe
  • Organizer: Equadiff 13
  • Place of Presentation: Charles University （Czech）
• [Presentation] バナッハ空間における非線形発展作用素と常微分方程式 (2013)
  • Author(s): 小林良和
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学
  • Invited
• [Presentation] 結晶粒界現象の放物型フェーズ・フィールドモデルにおける平滑化効果とエネルギー消散性 (2013)
  • Author(s): 白川健, 渡邉紘, S. Moll
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学
• [Presentation] 結晶粒界現象を記述する退化性を伴う1次元数学モデルに対する定性的性質 (2013)
  • Author(s): 渡邉紘, 白川健
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学
• [Presentation] 退化性を伴う1次元Kobayashi-Warren-Carterモデルに対する定性的性質 (2013)
  • Author(s): 渡邉紘, 白川健
  • Organizer: 第39回発展方程式研究会
  • Place of Presentation: 日本女子大学
• [Presentation] 強退化放物型方程式の初期値境界値問題に対する解の一意存在性 (2013)
  • Author(s): 渡邉 紘
  • Organizer: 第5回東北 楕円型放物型微分方程式研究集会
  • Place of Presentation: 東北大学
  • Invited
• [Presentation] Banach 空間上の Lipschitz 発展作用素 (2013)
  • Author(s): 冨澤佑季乃
  • Organizer: 日本数学会2013年度年会
  • Place of Presentation: 京都大学
• [Presentation] 強退化放物型方程式に対する動力学的接近 (2013)
  • Author(s): 渡邉 紘
  • Organizer: 日本数学会2013年度年会
  • Place of Presentation: 京都大学
• [Presentation] Strongly degenerate parabolic equations with discontinuous coefficients (2012)
  • Author(s): H. Watanabe
  • Organizer: Work Shop of JSPS-CNR Joint Research Project
  • Place of Presentation: 神戸大学(兵庫県)
  • Year and Date: 2012-08-03
• [Presentation] 増大度αの解析的半群の非線形摂動 (2012)
  • Author(s): 松本敏隆, 田中直樹
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学(東京都)
  • Year and Date: 2012-03-28
• [Presentation] 結晶粒界現象と関連する 1 次元放物型連立方程式に対するエネルギー消散解 (2012)
  • Author(s): 白川健, 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学(東京都)
  • Year and Date: 2012-03-28
• [Presentation] 強退化放物型方程式に対する初期値境界値問題の適切性 (2012)
  • Author(s): 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学(東京都)
  • Year and Date: 2012-03-28
• [Presentation] Global existence of unique solutions to nonautonomous differential equation in Banach spaces (2012)
  • Author(s): Y. Tomizawa, Y. Kobayashi and N. Tanaka
  • Organizer: Conference on Evolution Equations, Related Topics and Applications
  • Place of Presentation: 早稲田大学(東京都)
  • Year and Date: 2012-03-19
• [Presentation] 強退化放物型方程式に対する動力学的近似を用いたエントロピー解の構成 (2012)
  • Author(s): 渡邉紘
  • Organizer: 第38回発展方程式研究会
  • Place of Presentation: 日本女子大学(東京都)
• [Presentation] Nonautonomous differential equations and Lipschitz evolution operators in Banach spaces (2012)
  • Author(s): Y. Tomizawa, Y. Kobayashi and N. Tanaka
  • Organizer: International Conference on the Theory, Methods and Applications of Nonlinear Equations
  • Place of Presentation: Texas A&M University(アメリカ)
• [Presentation] Entropy solutions for strongly degenerate parabolic equations (2012)
  • Author(s): H. Watanabe
  • Organizer: 5th Polish-Japanese Days on Nonlinear Analysis in Interdisciplinary Sciences -Modelings, Theory and Simulations
  • Place of Presentation: 関西セミナーハウス(京都)
• [Presentation] 不連続な係数を持つ強退化放物型方程式の初期値問題に対するエントロピー解 (2012)
  • Author(s): 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学(福岡県)
• [Presentation] Unique solutions to nonautonomous differential equation in Banach spaces (2012)
  • Author(s): 冨澤佑季乃, 小林良和, 田中直樹
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学(福岡県)
• [Presentation] Nonautonomous differential equations in Banach spaces with an application (2012)
  • Author(s): Y. Kobayashi
  • Organizer: The 4th international Symposium on Banach and Function Spaces 2012
  • Place of Presentation: 九州工業大学(福岡県)
• [Presentation] Continuous dependence of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients (2012)
  • Author(s): H. Watanabe
  • Organizer: The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Orlando, Frorida ( アメリカ )
• [Presentation] Continuous dependence of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients (2012)
  • Author(s): Hiroshi Watanabe
  • Organizer: The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Orlando, Frorida（アメリカ）
• [Presentation] Strongly degenerate parabolic equations with discontinuous coefficients (2012)
  • Author(s): Hiroshi Watanabe
  • Organizer: Workshop of JSPS-CNR Joint Resarch Project
  • Place of Presentation: 神戸大学
  • Invited
• [Presentation] Nonautonomous differential equations in Banach spaces with an application (2012)
  • Author(s): Yoshikazu Kobayashi
  • Organizer: The Fourth International Symposium on BANACH and FUNCTION SPACES 2012
  • Place of Presentation: 九州工業大学
  • Invited
• [Presentation] Unique solutions to nonautonomous differential equations in Banach spaces (2012)
  • Author(s): 冨澤佑季乃
  • Organizer: 日本数学会2012年度秋季総合分科会
  • Place of Presentation: 九州大学
• [Presentation] 不連続な係数を持つ強退化放物型方程式の初期値問題に対するエントロピー解 (2012)
  • Author(s): 渡邉 紘
  • Organizer: 日本数学会2012年度秋季総合分科会
  • Place of Presentation: 九州大学
• [Presentation] Entropy solutions for strongly degenerate parabolic equations (2012)
  • Author(s): Hiroshi Watanabe
  • Organizer: 5th Polish-Japanese Days on Nonlinear Analysis in Interdisciplinary Science - Modelings, Theory and Simulations-
  • Place of Presentation: 関西セミナーハウス（京都）
  • Invited
• [Presentation] Nonautonomous differential equations and Lipschitz evolution operators in Banach spaces (2012)
  • Author(s): Yukino Tomizawa
  • Organizer: International Conference on the Theory, Methods and Applications of Nonlinear Equations
  • Place of Presentation: Texas A&M University（アメリカ）
• [Presentation] Lipschitz発展作用素と非自励微分方程式 (2012)
  • Author(s): 冨澤佑季乃
  • Organizer: 第38回発展方程式研究会
  • Place of Presentation: 日本女子大学
• [Presentation] 強退化放物型方程式に対する動力学的近似を用いたエントロピー解の構成 (2012)
  • Author(s): 渡邉 紘
  • Organizer: 第38回発展方程式研究会
  • Place of Presentation: 日本女子大学
• [Presentation] 増大度αの解析的半群の非線形摂動 (2012)
  • Author(s): 松本敏隆、田中直樹
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学(東京都）
• [Presentation] 結晶粒界現象と関連する1次元放物型連立方程式に対するエネルギー消散解 (2012)
  • Author(s): 白川 健、渡邉 紘
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学(東京都）
• [Presentation] 強退化放物型方程式に対する初期値境界値問題の適切性 (2012)
  • Author(s): 渡邉 紘
  • Organizer: 日本数学会(招待講演)
  • Place of Presentation: 東京理科大学(東京都）
• [Presentation] Global existence of unique solutions to nonautonomous differential equations in Banach spaces (2012)
  • Author(s): Y. Tomizawa, Y. Kobayashi and N. Tanaka
  • Organizer: CONFERENCE oN Evolution Equations, Related Topics and Applications
  • Place of Presentation: 早稲田大学(東京都）
• [Presentation] Solvability of boundary value problem for strongly degenerate parabolic equation with discontinuous coefficients (2011)
  • Author(s): H. Watanabe
  • Organizer: Dutch-Japanese Workshop, Analysis of non-equilibrium evolution problem : selected topics in material and life sciences
  • Place of Presentation: Eindhoven University of Technology ( オランダ )
  • Year and Date: 2011-11-08
• [Presentation] 結晶粒界の 1 次元Kobayashi-Waren-Carter モデルに対する弱解の存在 (2011)
  • Author(s): 白川健, 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 信州大学(長野県)
  • Year and Date: 2011-09-29
• [Presentation] 不連続な係数を持つ強退化放物型方程式に対する zero-flux 境界値問題 (2011)
  • Author(s): 渡邉紘
  • Organizer: 日本数学会
  • Place of Presentation: 信州大学(長野県)
  • Year and Date: 2011-09-29
• [Presentation] 結晶粒界の1次元Kobayashi-Warren-Carterモデルに対する弱解の存在 (2011)
  • Author(s): 白川 健、渡邉 紘
  • Organizer: 日本数学会
  • Place of Presentation: 信州大学（長野県）
• [Presentation] 不連続な係数を持つ強退化放物型方程式に対するzero-flux境界値問題 (2011)
  • Author(s): 渡邉 紘
  • Organizer: 日本数学会
  • Place of Presentation: 信州大学（長野県）
• [Presentation] Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients (2011)
  • Author(s): Hiroshi Watanabe
  • Organizer: DUTCH - JAPANESE WORKSHOP, Analysis of non-equilibrium evolution problems: selected topics in material and life sciences
  • Place of Presentation: Eindhoven University of Technology（オランダ）