On the asymptotics and well-posedness for Keller-Segel system of degenerate and singular type

• Project/Area Number: 15K04961
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Osaka University (2018); Kyushu University (2015-2017)
• Principal Investigator: Sugiyama Yoshie 大阪大学, 情報科学研究科, 教授 (60308210)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 退化放物型方程式 / Keller-Segel系 / 漸近解析 / 非自明定常解 / 退化型 / 閾値 / 定常解 / 有限伝播性 / Navier-Stokes方程式 / 退化放物型 / 特異放物型 / 初期値問題の時間局所適切性 / 爆発解 / Keller-Segel 系 / chemotaxis / well-posedness / Navier-Stokes 方程式 / Uniqueness

Outline of Final Research Achievements

We investigated asymptotic analysis at infinite time with the degenerate Keller-Segel system. More precisely, we proved the existence of nontrivial stationary solutions of the Keller-Segel system, and further proved the fact that the profiles for the Keller-Segel system, which are time-evolving solutions, have the same stationary solution at time infinity. The research results have been published in IUMJ in 2018 as a joint research with Mr. Jose Carrillo.

Academic Significance and Societal Importance of the Research Achievements

退化楕円型方程式の変分問題を解くことに帰着し，Keller-Segel系の非自明な定常解の存在証明を確立している．同問題の解析には，退化性が誘引する正則性欠如を克服する必要がある．我々は，球対称性を担保することで解のサポートコンパクト性を保証し，解析を容易にする手法を確立している．同手法は多様な退化型方程式系の解析に有効である．

Research Products

• [Int'l Joint Research] Imperial College London(英国)
• [Int'l Joint Research] Seoul National University(韓国)
• [Int'l Joint Research] Yonsei University(韓国)
• [Journal Article] Compactly supported stationary states of the degenerate Keller-Segel system in the diffusiondominated regime (2018)
  • Author(s): J.Carrillo and Y.Sugiyama
  • Journal Title: Indiana Univ. Math. J. Volume: 67
• [Journal Article] Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid (2016)
  • Author(s): H.Kozono, M.Miura and Y.Suiyama
  • Journal Title: J. Funct. Anal. Volume: 270 Pages: 1663-1683
  • Peer Reviewed
• [Journal Article] Uniqueness theorem on weak solutions to the Keller-Segel system of degenerate and singular types (2016)
  • Author(s): T.Kawakami and Y.Sugiyama
  • Journal Title: J. Differential Equations Volume: 260 Pages: 4683-4716
  • Peer Reviewed
• [Journal Article] Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid (2016)
  • Author(s): H.Kozono; M.Miurai; Y.Sugiyama
  • Journal Title: J. Funct. Anal. Volume: 270 Issue: 5 Pages: 1663-1683
  • DOI: 10.1016/j.jfa.2015.10.016
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Uniqueness theorem on weak solutions to the Keller-Segel system of degenerate and singular types (2016)
  • Author(s): T.Kawakami; Y.Sugiyama
  • Journal Title: J. Differential Equations Volume: 260 Issue: 5 Pages: 4683-4716
  • DOI: 10.1016/j.jde.2015.11.021
  • Peer Reviewed
• [Presentation] 脳動脈瘤治癒過程における血栓化ダイナミクスを説明する数理モデルの構築 (2018)
  • Author(s): 杉山由恵
  • Organizer: CREST・さきがけ・AIMaP合同シンポジウム「数学パワーが世界を変える2019」
  • Invited
• [Presentation] On the ε-regularity theorem for the Keller-Segel systems and its application to the analysis of singular sets (2018)
  • Author(s): 杉山由恵
  • Organizer: 京都大学NLPDEセミナー
  • Invited
• [Presentation] Measure valued solutions of 2D Keller-Segel systems (2018)
  • Author(s): 杉山由恵
  • Organizer: The 16th Linear and Nonlinear Waves
  • Int'l Joint Research / Invited
• [Presentation] Measure valued solutions of 3D Keller-Segel systems (2018)
  • Author(s): 杉山由恵
  • Organizer: RIMS 研究集会 「反応拡散方程式‐伝播現象と特異性の解析および諸科科学への応用」
  • Int'l Joint Research / Invited
• [Presentation] Compactly supported stationary states of the degenerate Keller-Segel system in the difusion-dominated regime (2018)
  • Author(s): 杉山由恵
  • Organizer: 2018年度偏微分方程式 集中セミナープログラム
  • Invited
• [Presentation] On the Structure of Solutions of Keller-Segel Systems with Sinks of Fluid (2018)
  • Author(s): 杉山由恵
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, SS10
  • Int'l Joint Research / Invited
• [Presentation] Compactly supported stationary states of the degenerate Keller-Segel systemin the difusion-dominated regime (2018)
  • Author(s): 杉山由恵
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, SS11
  • Int'l Joint Research / Invited
• [Presentation] 退化型Keller-Segel系の解構造について (2018)
  • Author(s): 杉山由恵
  • Organizer: 大阪大学数学科談話会
  • Invited
• [Presentation] On the structure of solutions of Keller-Segel systems with fluid and its application to life science (2018)
  • Author(s): 杉山由恵
  • Organizer: 大阪大学 微分方程式セミナー
  • Invited
• [Presentation] Compactly supported stationary states of the degenerate Keller-Segel systemin the difusion-dominated regime (2018)
  • Author(s): 杉山由恵
  • Organizer: 南大阪応用数学セミナー
  • Invited
• [Presentation] Global existence and finite time blow-up criterion of solutions to the Keller-Segel systems coupled with the Navier-Stokes fluid (2017)
  • Author(s): Yoshie Sugiyama
  • Organizer: Tianfu International Conference on PDEs
  • Int'l Joint Research / Invited
• [Presentation] Global existence and finite time blow-up criterion of solutions to the Keller-Segel systems coupled with the Navier-Stokes fluid (2017)
  • Author(s): Yoshie Sugiyama
  • Organizer: KWMS(Korean Women in Mathematical Sciences) International Conference
  • Int'l Joint Research / Invited
• [Presentation] On the structure of solutions of Keller-Segel systems with sinks of fluid (2017)
  • Author(s): Yoshie Sugiyama
  • Organizer: 第8回 拡散と移流の数理
  • Invited
• [Presentation] On the structure of solutions of Keller-Segel systems with sinks of fluid (2017)
  • Author(s): Yoshie Sugiyama
  • Organizer: The 3rd International Workshop on Mathematical Analysis of Chemotaxis
  • Int'l Joint Research / Invited
• [Presentation] Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid (2017)
  • Author(s): 三浦正成
  • Organizer: 日本数学会
  • Place of Presentation: 首都大学東京
• [Presentation] On H¨older continuity of solutions to fast diffusion equations with external forces and its applications (2017)
  • Author(s): 杉山由恵
  • Organizer: The second international workshop on mathematical analysis of chemotaxis
  • Place of Presentation: 東京理科大学
  • Int'l Joint Research / Invited
• [Presentation] Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with Navier-Stokes fluid (2016)
  • Author(s): 杉山由恵
  • Organizer: International Workshop on Mathematical Analysis of Chemotaxis
  • Place of Presentation: 東京理科大学
  • Year and Date: 2016-02-22
  • Int'l Joint Research / Invited
• [Presentation] Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with Navier-Stokes fluid (2016)
  • Author(s): 杉山由恵
  • Organizer: 北陸応用数理研究会2016 金沢大学
  • Place of Presentation: 金沢大学
  • Year and Date: 2016-02-17
  • Invited
• [Presentation] Global existence and finite time blow-up criterion of solutions to the Keller-Segel systems coupled with the Navier-Stokes fluid (2016)
  • Author(s): 杉山由恵
  • Organizer: Workshop on Nonlinear Partial Differential Equations -China-Japan Joint Project for Young Mathematicians 2016
  • Place of Presentation: East China Normal University, Shanghai
  • Int'l Joint Research / Invited
• [Presentation] Global existence and finite time blow-up criterion of solutions to the Keller-Segel systems coupled with the Navier-Stokes fluid (2016)
  • Author(s): 杉山由恵
  • Organizer: Interactions between Partial Differential Equations & Functional Inequalities
  • Place of Presentation: Intstitute Mittag-Leffler, Sweden
• [Presentation] Global existence and finite time blow-up criterion of solutions to the Keller-Segel systems coupled with the Navier-Stokes fluid (2016)
  • Author(s): 杉山由恵
  • Organizer: 7th Euro-Japanese Workshop on Blow-up
  • Place of Presentation: The Mathematical Research and Conference Center, Poland
• [Presentation] Mathematical Modeling and Analysis of Chemotactic Cell Migration (2015)
  • Author(s): 杉山由恵
  • Organizer: 京都大学 生命動態システム科学拠点セミナー
  • Place of Presentation: 京都大学
  • Year and Date: 2015-11-06
  • Invited