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Theory of global well-posedness on the nonlinear partial differential equations

Research Project

Project/Area Number 20224013
Research Category

Grant-in-Aid for Scientific Research (S)

Allocation TypeSingle-year Grants
Research Field Basic analysis
Research InstitutionWaseda University (2011-2012)
Tohoku University (2009-2010)

Principal Investigator

KOZONO Hideo  早稲田大学, 理工学術院, 教授 (00195728)

Co-Investigator(Kenkyū-buntansha) YANAGIDA Eiji  東京工業大学, 大学院・理工学研究科, 教授 (80174548)
ISHIGE Kazuhiro  東北大学, 大学院・理工学研究科, 教授 (90272020)
NAKAMURA Makoto  山形大学, 理学部, 教授 (70312634)
KUBO Hideo  北海道大学, 大学院・理学研究科, 教授 (50283346)
KANEDA Yukio  愛知工業大学, 工学部, 教授 (10107691)
ISHIHARA Takashi  名古屋大学, 大学院・工学研究科, 准教授 (10262495)
YOSHIMATSU Katsunori  名古屋大学, 大学院・工学研究科, 助教 (70377802)
KAGEI Yoshiyuki  九州大学, 大学院・数理学研究院, 教授 (80243913)
EI Shinichro  九州大学, マス・フォア・インダストリ, 教授 (30201362)
Project Period (FY) 2008 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥146,120,000 (Direct Cost: ¥112,400,000、Indirect Cost: ¥33,720,000)
Fiscal Year 2012: ¥40,820,000 (Direct Cost: ¥31,400,000、Indirect Cost: ¥9,420,000)
Fiscal Year 2011: ¥35,360,000 (Direct Cost: ¥27,200,000、Indirect Cost: ¥8,160,000)
Fiscal Year 2010: ¥35,360,000 (Direct Cost: ¥27,200,000、Indirect Cost: ¥8,160,000)
Fiscal Year 2009: ¥34,580,000 (Direct Cost: ¥26,600,000、Indirect Cost: ¥7,980,000)
Keywords非線形解析学 / 偏微分方程式論 / 調和解析学 / 関数解析学 / Keller-Segel 方程式系 / Navier-Stokes 方程式 / 弱解の一意性 / 時間大域的弱解 / L^p-L^r-型評価 / 最大正則性定理 / 関数方程式 / 非線形偏微分方程式 / keller-Segel系 / タイプI型爆発解 / タイプII型爆発解 / 自己相似解 / 局所存在定理 / Navier-Stokes方程式 / 強エネルギー不等式 / 非斉次境界値問題
Research Abstract

We investigate the local existence of strong solutions and their blow-up within a finite time in arbitrary dimensional domains. The life-span of local solutions is characterized in terms of the L^1 and L^p-norms of the given initial data. Simultaneously, it is clarified that the total mass and the second momentum of the initial data together with the coefficient of the system of equations have a great influence on the blow-up phenomena. As an application, we prove that the blow-up solution either exhibits a definite blow-up rate determined by p, or oscillates in L^1 with the larger amplitude than the absolute constant. Furthermore, in multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified.

Assessment Rating
Verification Result (Rating)

A

Report

(5 results)
  • 2013 Research Progress Assessment (Verification Result) ( PDF )
  • 2012 Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report   Self-evaluation Report ( PDF )
  • Research Products

    (36 results)

All 2012 2011 2010 2009 2008 Other

All Journal Article (21 results) (of which Peer Reviewed: 20 results) Presentation (13 results) (of which Invited: 3 results) Remarks (2 results)

  • [Journal Article] Existence and uniqueness theorem on mild solutions to the Keller-Segel system2012

    • Author(s)
      KOZONO,H., Sugiyama, Y., Wachi, T.,
    • Journal Title

      J. Differential Equations

      Volume: 252 Issue: 2 Pages: 1213-1228

    • DOI

      10.1016/j.jde.2011.08.025

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the stationary Navier-Stokes flows around a rotating body2012

    • Author(s)
      Heck, H., Kim, H., Kozono, H.,
    • Journal Title

      Manuscripta Math.

      Volume: 138 Issue: 3-4 Pages: 315-345

    • DOI

      10.1007/s00229-011-0494-1

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and uniqueness theorem on weak solutions to the parabolic-elliptic Keller-Segel system2012

    • Author(s)
      Kozono,H., Sugiyama,Y., Yahagi,Y.,
    • Journal Title

      J. Differential Equations

      Volume: 253 Issue: 7 Pages: 2295-2313

    • DOI

      10.1016/j.jde.2012.06.001

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence2011

    • Author(s)
      Farwig,R.,Kozono,H., Sohr,H.
    • Journal Title

      Rend. Semin. Mat. Univ. Padova

      Volume: 125 Pages: 51-70

    • DOI

      10.4171/rsmup/125-4

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Non-existence of finite-time self-similar solutions of the Keller-Segel system in the scaling invariant class2010

    • Author(s)
      Kozono,H.,Sugiyama,Y.,Takada.R
    • Journal Title

      J. Math. Anal. Appl.

      Volume: 365 Issue: 1 Pages: 60-66

    • DOI

      10.1016/j.jmaa.2009.09.063

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Blow-up set for a semilinear heat equation with small diffusion2010

    • Author(s)
      Y.Fujishima, K.Ishige
    • Journal Title

      J.Differential Equations 249

      Pages: 1056-1077

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Parabolic quasi-concavity for solutions to parabolic problems in convex rings2010

    • Author(s)
      K.Ishige, P.Salani
    • Journal Title

      Math.Nachr.(雑誌Math.Nachr.のEditor Choice を受賞) 283

      Pages: 1526-1548

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Non-existence of finite-time self-similar solutions of the Keller-Segel system in the scaling invariant class2010

    • Author(s)
      小薗英雄・杉山由恵・高田了
    • Journal Title

      J.Math.Anal.Appl.

      Volume: 365巻 Pages: 60-66

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Strong solutions to the Keller-Segel system with the weak $L^{¥frac{n}{2}}$ initial data and its application to the blow-up rate2010

    • Author(s)
      小薗英雄・杉山由恵
    • Journal Title

      Math.Nach.

      Volume: 283巻 Pages: 732-751

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global weak solutions of the Navier-Stokes system with nonzero boundary conditions2010

    • Author(s)
      小薗英雄・R.Farwig,・H.Sohr
    • Journal Title

      Funkcialaj Ekvacioj

      Volume: 51巻 Pages: 231-247

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Nonhomogeneous boundary value problems for stationary Navier-Stokes equations in a bounded domain with a multiply connected boundary2009

    • Author(s)
      Kozono,H.,Yanagisawa,T
    • Journal Title

      Pacific Math. J.

      Volume: 243 Issue: 1 Pages: 127-150

    • DOI

      10.2140/pjm.2009.243.127

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Leray's problem on the stationary Navier-Stokes equations with inhomogeneous boundary data2009

    • Author(s)
      Kozono,H.,Yanagisawa,T
    • Journal Title

      Math. Z.

      Volume: 262 Issue: 1 Pages: 27-39

    • DOI

      10.1007/s00209-008-0361-2

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Global DIV-CURL Lemma in bounded domains in $R^3$2009

    • Author(s)
      Kozono,H.,Yanagisawa,T
    • Journal Title

      Jour. Funct. Anal.

      Volume: 256 Issue: 11 Pages: 3847-3859

    • DOI

      10.1016/j.jfa.2009.01.010

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces2009

    • Author(s)
      Kozono,H., Sugiyama,Y
    • Journal Title

      J. Differential Equations

      Volume: 247 Issue: 1 Pages: 1-32

    • DOI

      10.1016/j.jde.2009.03.027

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] $L^r$-variational inequality for vector Fields and the Helmholtz-Weyl decomposition in bounded domains2009

    • Author(s)
      Kozono,H.,Yanagisawa,T
    • Journal Title

      Indiana Univ. Math. J.

      Volume: 58 Issue: 4 Pages: 1853-1920

    • DOI

      10.1512/iumj.2009.58.3605

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Leray's problem on the stationary Navier-Stokes equations with inhomogeneous boundary data2009

    • Author(s)
      Kozono, H., Yanagisawa, T.
    • Journal Title

      Math.Z. 262

      Pages: 27-39

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Global DIV-CURL Lemma in bounded domains in R^32009

    • Author(s)
      Kozono, H, Yanagisawa, T
    • Journal Title

      Jour.Funct. Anal 256

      Pages: 847-3859

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] L^r-variational inequality for vector fields and the Helmholtz-Weyl decomposition in boundeddomains2009

    • Author(s)
      Kozono, H., Yanagisawa, T.
    • Journal Title

      Indiana Univ.Math.J. 58

      Pages: 1853-1920

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ and its application to the self-similar solution2008

    • Author(s)
      Kozono,H., Sugiyama,Y
    • Journal Title

      Indiana Univ. Math. J.

      Volume: 57 Issue: 4 Pages: 1467-1500

    • DOI

      10.1512/iumj.2008.57.3316

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Local existence and finite time blow-up in the 2-D Keller-Segel system2008

    • Author(s)
      Kozono,H., Sugiyama,Y.
    • Journal Title

      J. Evol. Equ.

      Volume: 8 Issue: 2 Pages: 353-378

    • DOI

      10.1007/s00028-008-0375-6

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO,2008

    • Author(s)
      Kozono,H.,Wadade,H
    • Journal Title

      Math. Z.

      Volume: 259 Issue: 4 Pages: 953-950

    • DOI

      10.1007/s00209-007-0258-5

    • Related Report
      2012 Final Research Report
  • [Presentation] Uniqueness criterion of weak solutions to the Navier-Stokes equations in general unbounded domains2012

    • Author(s)
      Hideo KOZONO
    • Organizer
      POSTECH Summer Workshop on Application and Analysis on PDEs
    • Place of Presentation
      Postech, Pohang, Korea
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Uniqueness of weak solutions to the Navier-Stokes equations in general unbounded domains2012

    • Author(s)
      Hideo KOZONO
    • Organizer
      Workshop on Complex Fluids
    • Place of Presentation
      TU Darmstadt, Germany
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems2011

    • Author(s)
      Hideo KOZONO
    • Organizer
      VORTICITY, ROTATION AND SYMMETRY (II) , REGULARITY OF FLUID MOTION
    • Place of Presentation
      Luminy, France
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Blow-up for a semilinear parabolic equation with large diffusion on R^N2010

    • Author(s)
      石毛和弘
    • Organizer
      4^<th> Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      Lorentz Center, Leiden
    • Year and Date
      2010-09-10
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Global compensated compactness theorem for general differential operators of first order.2010

    • Author(s)
      小薗英雄
    • Organizer
      The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Dresden, Germany
    • Related Report
      2012 Final Research Report
  • [Presentation] Leray's inequality in 3D multi-connected domains2009

    • Author(s)
      小薗英雄
    • Organizer
      Workshop on Mathematical Aspects of Hydrodynamics
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany
    • Related Report
      2012 Final Research Report
  • [Presentation] Leray's inequality in 3D domains2009

    • Author(s)
      小薗英雄
    • Organizer
      International Conference on Parabolic Equations 2009
    • Place of Presentation
      Banach Center, Bedlewo, Poland
    • Related Report
      2012 Final Research Report
  • [Presentation] 領域の位相不変量と定常ナビエ・ストークス方程式の可解性について2009

    • Author(s)
      小薗英雄
    • Organizer
      中央大学 第49回 Encounter with Mathematics 流体基礎方程式 いろいろな視点から見た流体方程式」
    • Place of Presentation
      中央大学
    • Related Report
      2012 Final Research Report
  • [Presentation] Leray's inequality in 3D multi-connected domains2009

    • Author(s)
      小薗英雄
    • Organizer
      Workshop on Mathematical Aspects of Hydrodynamics
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Ger many
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Global Div-Curl Lemma in 3D bounded domains2009

    • Author(s)
      小薗英雄
    • Organizer
      International Conference on HarmonicAnalysis and Partial Differential Equations with Applications
    • Place of Presentation
      Beijing Normal University(Beijing, China)
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Leray's inequality in 3D domains2009

    • Author(s)
      小薗英雄
    • Organizer
      International Conference on Parabolic Equations2009 : On the occasion of Prof.Herbert Amann's 70h birthday
    • Place of Presentation
      Banach Center(Bedlewo, Poland)
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Global Div-Curl Lemma in 3D bounded domains2009

    • Author(s)
      小薗英雄
    • Organizer
      German-Japanese International Trainning Groups Waseda-Darmstadt University Mathematical Fluid Dynamics Launching Workshop
    • Place of Presentation
      早稲田大学
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Global Div-Curl Lemma on bounded domains in $R^3$.2008

    • Author(s)
      小薗英雄
    • Organizer
      Vorticity, Rotation and Symmetry-Stabilizing and Destabilizing Fluid Motion C.I.R.M.
    • Place of Presentation
      Luminy, France
    • Related Report
      2012 Final Research Report
  • [Remarks]

    • URL

      http://www.math.tohoku.ac.jp/researchfields/kozono.html

    • Related Report
      2012 Final Research Report
  • [Remarks] ホームページ

    • URL

      http://www.math.tohoku.ac.jp/researchfields/kozono.html

    • Related Report
      2010 Self-evaluation Report

URL: 

Published: 2009-04-01   Modified: 2019-07-29  

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