• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Research on a refinement of the energy inequality for weak solutions to the Navier-Stokes equations

Research Project

Project/Area Number 12640200
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTOHOKU UNIVERSITY

Principal Investigator

NAGASAWA Takeyuki  Tohoku University, Math. Inst., Associate Professor, 大学院・理学研究科, 助教授 (70202223)

Co-Investigator(Kenkyū-buntansha) TSUTSUMI Yoshio  Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (10180027)
TAKAGI Izumi  Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (40154744)
SHIMAKURA Norio  Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (60025393)
FUJIIE Seturo  Tohoku University, Math. Inst., Lecturer, 大学院・理学研究科, 講師 (00238536)
KOZONO Hideo  Tohoku University, Math. Inst., Professor, 大学院・理学研究科, 教授 (00195728)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2000: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsNavier-Stokes equations / Energy identity / Energy inequality / Weak solution / Regurality / Uniqueness
Research Abstract

Though the mathematical research on the Navier-Stokes equations that describe the motion of incompressible fluid has a long history, the theory has not completed yet. In particular concerning the problems of the regularity and uniqueness of weak solutions, we have only partial answers. A weak solution satisfies equations only in some weak sense, and therefore it may have singular points. If a solution is smooth, then it preserves energy with respect to time variable. For some weak solutions we can show the non-increasing property of energy, but it is uncertain whether they preserve energy or not. It is called the "energy inequality." For weak solutions the integrability of time-derivative is unclear only from the definition. This is why we cannot show the preservation of energy. This fact suggests that time-derivative is a singular measure with respect to "time". Consequently we must consider the integral of this "singular" part to show the preservation of energy.
We know that it is possible to estimate the decrease of energy from below by use of fractional time-derivative for the weak solution constructed by the method of discrete Morse flow. This is a new estimate. The purpose of this research is to study the possibility of such a refinement for any weak solution. For precise, we clarified the following facts. We consider weak solutions as functions which map to the space of square-integrable functions. Then the decrease of energy is related to the limit of the 1/2-time-difference of solutions in the sense of Nikol'skii. If we assume that the limit is zero with or without some speed, then we can show the energy identity with an additional term which compensates the decrease of energy. Furthermore without the assumption of the existence of limit, we can show the energy identity with another additional term of different expression. The difference of expression comes from that of topology of convergence of limit of time-difference.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (72 results)

All Other

All Publications (72 results)

  • [Publications] T.Nagasawa: "Closed surfaces minimizing the bending energy under prescribed area and volume"International Conference on Differential Equations Berlin 1999(ed. : F. Fiedler, K. Groger, J. Sprekels). 1. 561-563 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Nagasawa: "Blow-up solutions for ordinary differential equations associated to harmonic maps and their applications"J. Math. Soc. Japan. 53・2. 485-500 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Nagasawa: "A new energy inequality and partial regularity for weak solutions of Navier-Stokes equations"J. Math. Fluid Mech.. 3・1. 40-56 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Nagasawa: "A refinement of the energy inequality for the Navier-Stokes equations"Nonlinear Anal.. 47・6. 4245-4256 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Nakane: "Numerical analysis for hyperbolic Ginzburg Landau system"Nonlinear Anal.. (出版予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Nagasawa: "Bifurcating critical points of bending energy under constraints related to the shape of red blood cells"Calc. Var. Partial Differential Equations. (出版予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N.Shimakura: "Normal coordinate systems from a view point of real analysis"Tohoku Math. J.(2). 52・4. 533-553 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W.M.Ni: "Stability of least energy patterns of the shadow system for an activator-inhibitor model"Japan J. Indust. Appl. Math.. 18・2. 259-272 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.-S.Lin: "Method of rotating planes applied to a singularly perturbed Neumann problem"Calc. Var. Partial Differential Equations. 13・4. 519-536 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Ozawa: "On the coupled system of nonlinear wave equations with different propagation speeds"Evolution Equations : Existence, Regularity and Singularities Banach Center Publications. 52. 181-188 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Nakanishi: "Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrodinger equation"Proceedings of IMS Conference on Differential Equations from Mechanics. (出版予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 堤 誉志雄: "非線形波動方程式の解の大域存在と爆発"数学. 53・2. 139-156 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kozono: "Bilinear esitimates in BMO and the Navier-Stokes equations"Math. Z.. 235・1. 173-194 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kozono: "Limiting case of the Sobolev inequality in BMO, with application to the Euler equations"Commun. Math. Phy.. 214・1. 191-200 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kozono: "Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations"J. Funct. Anal.. 176・2. 153-197 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kozono: "On well-posedness of the Navier-Stokes equations Mathematical fluid mechanics"Adv. Math. Fluid Mech.. 207-236 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kozono: "Well-posedness for the Benjamin equations"J. Korean Math. Soc.. 38・6. 1205-1234 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kozono: "Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains"Math. Ann.. 320・4. 709-730 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kozono: "Weak solutions of the Navier-Stokes equations with test functions in the weak-L^n space"Tohoku Math. J.(2). 53・1. 55-79 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Fujiie: "Semiclassical phase shift and time delay at a barrier top for the radial Schrodinger equation"Vietnam SAMM monograph. (出版予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Fujiie: "Semiclassical behavior of the scattering phase near a critical value of the potential"京都大学数理解析研究所講究録. 1212. 18-31 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 佐藤 得志: "測度を含む半線形楕円型方程式の正値解について(On positive solutions to some semilinear elliptic equations involving finite Radon measures)"京都大学数理解析研究所講究録. 1204. 34-49 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Nagasawa: "Closed surfaces minimizing the bending energy under prescribed area and volume"International conference on Differential Eequations Berlin 1999 (ed, : F. Fiedler, K. Groger, J. Sprekels). 1. 561-563 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Nagasawa: "Blow-up solutions for ordinary differential equations associated to harmonic maps and their applications"J. Math. Soc. Japan. 53 (2). 485-500 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Nagasawa: "A new energy inequality and partial regularity for weak solutions of Navier-Stokes equations"J. Math. Fluid Mech.. 3(1). 40-56 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Nagasawa: "A refinement of the energy inequality for the Navier-Stokes equations"Nonlinear Anal.. 47 (6). 4245-4246 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Nakane: "Numerical analysis for hyperbolic Ginzburg Landau system"Nonlinear Anal.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Nagasawa: "Bifurcating critical points of bending energy under constraints related to the shape of red blood cells"Calc. Var. Partial Differential Equations. (to apper).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N. Shimakura: "Normal coordinate systems from a view point of real analysis"Tohoku Math. J. (2). 52 (4). 533-553 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] W.-M. Ni: "Stability of least energy patterns of the shadow system for an activator-inhibitor model"18 (2). 259-272 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] C.-S. Lin: "Method of rotating planes applied to a singularly perturbed Neumann problem"Calc. Var. Partial Differential Equations. 13 (4). 519-536 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Ozawa: "On the coupled system of nonlinear wave equations with dirrerent propagation speeds"Evolution Equations : Existence, Regularity and Shingularities Banach Center Publications. 52. 181-188 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Nakanishi: "Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrodinger equation"Proseedings of IMS Conference on Differential Equations from Mechanics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Tsutsumi: "Global existence and blow-up of solutions to nonlinear wave equations (in Japanese)"Sugaku. 53 (2). 139-156 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kozono: "Bilinear estimates in BMO and the Navier-Stokes equations"Math. Z.. 235 (1). 173-194 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kozono: "Limiting case of the Sobolev inequality in BMO, with application to the Euler equations"Commun. Math. Phy.. 214 (1). 191-200 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kozono: "Asymptotic satability of large solutions with large perturbation to the Navier-Stokes equations"J. Funct. Anal.. 176 (2). 153-197 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kozono: "On well-posedness of the Navier-Stokes equations Mathematical fluid mechanics"Adv. Math. Fluid Mech.. 207-236 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kozono: "Well-posedness for the Benjamin equations"J. Korean Math. Soc.. 38 (6). 1205-1234 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kozono: "Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains"Math. Ann.. 320 (4). 709-730 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Konozo: "Weak solution of the Navier-Stokes equations with test functions in the weak-L^n space"Tohoku Math. J. (2). 53 (1). 55-79 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Fujiie: "Semiclassical phase shift and time delay at a barrier top for the radial Schrodinger equation"Vietnam SAMM monograph. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Fujiie: "Semiclassical behavior of the scattering phase near a critical value of the potential"Surikaisekikenkyusho Kokyuroku. 1212. 18-31 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Sato: "On positive solutions to some semilinear elliptic equations involving finite Radon measures (in Japanese)"Surikaisekikenkyusho Kokyuroku. 1204. 34-49 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Nagasawa: "Blow-up solutions for ordinary differential equations associated to harmonic maps and their applications"J.Math. Soc. Japan. 53・2. 485-500 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Nagasawa: "A new energy inequality and partial regularity for weak solutions of Navier-Stokes equations"J.Math. Fluid Mech.. 3・1. 40-56 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Nagasawa: "A refinement of the energy inequality for the Navier-Stokes equations"Nonlinear Anal. 47・6. 4245-4256 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Nakane: "Numerical analyis for hyperbolic Ginzburg Landau system"Nonlinear Anal. (出版予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Nagasawa: "Bifurcating critical points of bending energy under constraintsrelated to the shape of red blood cells"Calc. Var. Partial Differential Equations. (出版予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] W.M.Ni: "Stability of least energy patterns of the shadow system for an activator inhibitor model"Japan J. Indust. Appl. Math. 18・2. 259-272 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] C.S.Lin: "Method of rotating planes applied to a singularly perturbed Neumann problem"Calc. Var. Partial Differential Equations. 13・4. 519-536 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Nakanishi: "Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrodinger equation"Proceedings of IMS Conference on Differential Equations from Mechanics. (出版予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 堤 誉志雄: "非線形波動方程式の解の大域存在と爆発"数学. 53・2. 139-156 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kozono: "On well-posedness of the Navier-Stokes equations Mathematical fluid mechanics"Adv. Math. Fluid Mech.. 207-236 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kozono: "Well-posedness for the Benjamin equations"Korean Math. Soc.. 38・6. 1205-1234 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kozono: "Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains"Math. Ann.. 320・4. 709-730 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kozono: "Weak solutions of the Navier-Stokes equations with test functions in the weak-L^n space"Tohoku Math.J. (2). 53・1. 55-79 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Fujiie: "Semiclassical phase shift and time delay at a barrier top for the radial Schrodinger equation"Vietnam SAMM monograph.

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Fujiie: "Semiclassical behavior of the scattering phase near a critical value of the potential"京都大学数理解析研究所講究録. 1212. 18-31 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 佐藤 得志: "測度を含む半線形楕円型方程式の正値解について(On positive solutions to some semilinear elliptic equations involving finite Radon measures)"京都大学数理解析研究所講究録. 1204. 34-49 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Nagasawa: "Closed surfaces minimizing the bending energy under prescribed area and volume"International Conference on Differential Equations Berlin 1999(ed.: F.Fiedler, K.Groger, J.Sprekels). 1. 561-563 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Nagasawa: "Blow-up solutions for ordinary differential equations associated to harmonic maps and their applications"J.Math.Soc.Japan. 53(出版予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Nagasawa: "A new energy inequality and partial regularity for weak solutions of Navier-Stokes equations"J.Math Fluid Mech.. 3(出版予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Nagasawa: "A refinement of the energy inequality for the Navier-Stokes equations"Nonlinear Anal.. (出版予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] N.Shimakura: "Normal coordinate systems from a view point of real analysis"Tohoku Math.J.. 52・4. 533-553 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] W.-M.Ni: "Stability of least energy patterns of the shadow system for an activator-inhibitor model"Japan J.Indust.Appl.Math.. 18・2(出版予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] C.-S.Lin: "Method of rotating planes applied to a singularly perturbed Neumann problem"Calc.Var.Partial Differential Equations. (出版予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Ozawa: "On the coupled system of nonlinear wave equations with different propagation speeds"Evolution Equations : Existence, Regularity and Singularities Banach Center Publications. 52. 181-188 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Kozono: "Bilinear esitimates in BMO and the Navier-Stokes equations"Math.Z.. 235・1. 173-194 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H,Kozono: "Limiting case of the Sobolev inequality in BMO, with application to the Euler equations"Commun.Math.Phy... 214・1. 191-200 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Kozono: "Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations"J.Func.Anal.. 176・2. 153-197 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 佐藤得志: "測度を含む半線型楕円型方程式の正値解について"京都大学数理科学研究所講究録. (出版予定).

    • Related Report
      2000 Annual Research Report

URL: 

Published: 2000-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi