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2000 Fiscal Year Final Research Report Summary

Research on well-posedness for the Navier-Stokes equations

Research Project

Project/Area Number 09440056
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionTohoku University (1999-2000)
Nagoya University (1997-1998)

Principal Investigator

KOZONO Hideo  Tohoku University, Graduate School of Science, Prof., 大学院・理学研究科, 教授 (00195728)

Co-Investigator(Kenkyū-buntansha) NAGASAWA Takeyuki  Tohoku University, Graduate School of Science, Ass.Prof., 大学院・理学研究科, 助教授 (70202223)
TSUTSUMI Yoshio  Tohoku University, Graduate School of Science, Prof., 大学院・理学研究科, 教授 (10180027)
TAKAGI Izumi  Tohoku University, Graduate School of Science, Prof., 大学院・理学研究科, 教授 (40154744)
TACHIZAWA Kazuya  Tohoku University, Graduate School of Science, Lect., 大学院・理学研究科, 講師 (80227090)
CHIHARA Hiroyuki  Tohoku University, Graduate School of Science, Ass.Prof., 大学院・理学研究科, 助教授 (70273068)
Project Period (FY) 1997 – 2000
KeywordsNavier-Stokes Equations / Sobolur Space / Inter polation space / Fowwer transform / singular integral operation / Stoker operator / Lorentz space / exterior problem
Research Abstract

In a domain Ω⊂R^n, consider a weak solution u of the Navier-Stokes equations in the class u∈L^∞ (0, T ; L^n (Ω)). If lim sup_<t-t_*-0>‖u (t) ‖^n_n-‖u (t_*) ‖^n_n is small at each point of t_*∈ (0, T), then u is regular on Ω^^-× (0, T). As an application, we give a precise characterization of the singular time, i.e., we show that if a solution u of the Navier-Stokes equations is initially smooth and loses its regularity at some later time T_*<T, then either lim sup_<t-T_*-0>‖u (t) ‖_<L^n (Ω) >= +∞, or u (t) oscillates in L^n (Ω) around the weak limit w-lim_<t-T_*-0>u (t) with sufficiently large amplitude. Furthermore, we prove that every weak solution u of bounded variation on (0, T) with values in L^n (Ω) becomes regular.
Consider the nonstationary Navier-Stokes equations in Ω× (0, T), where Ω is a domain in R^3. We show that there is an absolute constant ε_0 such that every weak solution u with the property sup_<t∈ (a, b) >‖u (t) ‖^3_W (D) 【less than or equal】ε_0 is necessarily of class C^∞ in the space-time variables on any compact subset of D× (a, b), where D ⊂⊂Ω and 0<a<b<T.As an application, we prove that if the weak solution u behaves around (x_0, t_0) ∈Ω× (0, T) like u (x, t) =o (|x-x_0|^<-1>) as x→x_0 uniformly in t in some neighborhood of t_0, then (x_0, t_0) is a removable singularity of u.
Consider weak solutions w of the Navier-Stokes equations in Serrin's class
w∈L^α (0, ∞ ; L^q (Ω)) for 2/α + 3/q = 1 with 3<q【less than or equal】∞,
where Ω is a general unbounded domain in R^3. We shall show that although the inital and exteral disturbances from w are large, every perturbed flow u with the energy inequality converges asymptotically to w as
‖υ (t) -w (t) ‖_<L^2 (Ω) >→0, ‖▽υ(t) -▽w (t) ‖_<L^2 (Ω) >=O (t^<-1/2>) as t→∞.

  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] KOZONO,H.,YAMAZAKI,M: "Exterior Problem for the Stationary Navia-Stokes equations in the Lo rant-space"Math.Amn.. 310. 279-305 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KOZONO,H.,YAMAZAKI,M: "On a larger class of Stable Solutions to the Navia-Stokes equations in exterive domanins"Math.Z.. 228. 751-785 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KOZONO,H.,: "Removable singularities of the weak Solutions to the Navia-Stoke equations"Comman Pastoal Diff.Eq.. 23. 949-966 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KOZONO,H.: "L'-Solutions to the Navia-Stokes equations the exterior domain"Math.Ann.. 312. 319-340 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KoZoNo,H.,Taniuchi,Y: "Bilinear estimates in BMO and the Navia-Stokes equations"Math.Z.. 235. 173-194 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KoZoNo.H.: "Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equation"J.Func.Anal.. 176. 153-197 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] KOZONO,H.,Shibuta,Y.: "Recent topicts on Mathematical Theory of viscom Incompressible Fluid"紀伊国屋書店. 270 (1988)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kozono, H., Yamazaki, M.: "Exterior problem for the stationary Navier-Stokes equations in the Lorentz space"Math.Ann.. 310. 279-305 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kozono, H., Yamazaki, M.: "On a larger class of stable solutions to the Navier-Stokes equations in exterior domains"Math.Z.. 228. 751-785 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kozono, H.: "Removable singularities of weak solutions to the Navier-Stokes equations"Communications in Partial Differential Equations. 23. 949-966 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kozono, H.: "L^1-solutions of the Navier-Stokes equations in exterior domains"Math.Ann.. 312. 319-340 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kozono, H., Taniuchi, Y.: "Bilinear esitimates in BMO and the Navier-Stokes equations"Math.Z.. 235. 173-194 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kozono, H.: "Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations"J.Func.Anal.. 176. 153-197 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kozono, H., Shibata, Y.: "Recent Topics on Mathematical Theory of Viscous Incompressible Fluid, Lecture Notes in Numerical and Applied Analysis"Kinokuniya (Vol.16). (1998)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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