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Study of the stability of solutions to some nonlinear evolution equations based on recent development of real analysis

Research Project

Project/Area Number 15340204
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionWaseda University

Principal Investigator

SHIBATA Yoshihiro  Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (50114088)

Co-Investigator(Kenkyū-buntansha) TANAKA Kazunaga  Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (20188288)
YAMAZAKI Masao  Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (20174659)
KOBAYASHI Takayuki  Saga Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (50272133)
SHIMIZU Senjo  Shizuoka Univ., Faculty of Engineering, Assistant Professor, 工学部, 助教授 (50273165)
HISHIDA Toshiaki  Nigata Univ., Faculty of Engineering, Assistant Professor, 工学部, 助教授 (60257243)
Project Period (FY) 2003 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥12,000,000 (Direct Cost: ¥12,000,000)
Fiscal Year 2006: ¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2005: ¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2004: ¥4,000,000 (Direct Cost: ¥4,000,000)
KeywordsStokes equation / Neumann problem / Maximal regularity / Navier-Stokes equation / free boundary problem / flow past rotating body / perturbed half-space / Oseen equation / 数学 / 基礎解析学
Research Abstract

We study the spectral analysis of Stokes equations based on the recent development of the real analysis, Fourier analysis and functional analysis and its application to the Naveir-Stokes equations in several different situations arising from the mathematical physics.
1) We studied an inncompressible viscous flow past a rigid body, which is mathematically described by Oseen equations. We studied the decay properties of the Oseen semigroup in the exterior domain and showed global in time stability Navier-Stokes flow past a rigid body
2) We studied an inncompressible viscous flow in a perturbed half-space which describes for example flow past high buildings. We studied an optimal decay properties of solutions to the Stokes equations in a perturbed half-space and proved a global in time unique existence of solutions to the Navier-Stokes equations in a perturbed half-space with small initial data.
3) We proved the maximal regularity of solutions to the Stokes equation with the Neumann boundry … More condition in a bonded domain. We use some recent development of the operator-valued Fourier analysis by Weis and Denk-Hieber-Pruss. Our method is very simple compared with previous results and seems to be applicable to linear evoulution equations of parabolic type. Moreover, we proved local in time unique existence of strong solutions with arbitrary initial data and global in time unique existence of strong solutions with some small initial data of the free boundary problem of Navier Stokes equations which describes the transient motion of an isolated volume of viscous incompressible fluid
4) We studied an inncompressible viscous flow past a rotating rigid body. This problem was already studied by Galdi and Galdi and Silvestre in the L_2 framework. Our main contribution is to show the decay estimate of the continuous semigroup associated with linearized problem. The main difficulty comes from first order differential system with polynomially growing coefficients which is not subordinated by the Laplacian. We developed new technique to investigate the high frequency part of the spectrum. Less

Report

(5 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (54 results)

All 2007 2006 2005 2004 2003 Other

All Journal Article (46 results) Book (1 results) Publications (7 results)

  • [Journal Article] Decay properties of the Stokes semigroup in exterior domains with Neumann condition2007

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      J.Math.Soc.Japan 59

      Pages: 1-35

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Decay estimates of the Stokes flow around a rotating obstacle,2007

    • Author(s)
      T.Hishida, Y.Shibata
    • Journal Title

      RIMS Kokyuroku Bessatsu, (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Decay properties of the Stokes semigroup in exterior domains with Neumann condition,2007

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Journal of Mathematical Society of Japan 59

      Pages: 1-35

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid2007

    • Author(s)
      Y.Shibata, K.Tanaka
    • Journal Title

      Computers & Mathematics with applications

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Globally in time existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle2006

    • Author(s)
      T.Hishida, Y.Shibata
    • Journal Title

      WAEAS Trans. Math. Profeedings of WAEAS (Conf on Continuum Mechanics) 5

      Pages: 303-307

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Globally in time existence theorem for the Navier-Stokes flow in the exterior of a rotating bostacle2006

    • Author(s)
      T.Hishida, Y.Shibata
    • Journal Title

      WSEAS Trans.Math.5, "Proceedings of WAEAS International Conference on Continuum Mech.

      Pages: 303-307

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A Stokes approximation of two dimensional exterior Oseen flow near the boundary,2006

    • Author(s)
      M.Okamura, Y.Shibata, N.Yamaguchi
    • Journal Title

      Proceedings of MSJ-IRI 005 "Asymptotic Analysis and Singularity" Advanced Studies in Pure Mathematics, (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Global existence of strong solution to the micropolar fluid system,2006

    • Author(s)
      Y.Shibata, N.Yamaguchi
    • Journal Title

      Hyperbolic Problems, Theory, Numerics and Applications II(F.Asakura et al.ed.)(Yokohama Publishers,)

      Pages: 305-312

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] L_p-L_q estimate of the Oseen semigroup in an n-dimensional extdrior domain2005

    • Author(s)
      Y.Enomoto, Y.Shibata
    • Journal Title

      J. math. fluid mech. 7

      Pages: 339-367

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the Stokes and Navier-Stokes equation in a perturbed half-space2005

    • Author(s)
      T.Kubo, Y.Shibata
    • Journal Title

      Advances in Differential Equations 10

      Pages: 695-720

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] Uniform estimates in the velocity at infinity for stationary solutions to the Navier-Stokes exterior problem2005

    • Author(s)
      Y.Shibata, M.Yamazaki
    • Journal Title

      Japanese J. Math. (New Series) 31

      Pages: 225-279

    • NAID

      10020427306

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] L_p-L_q estimate of the Oseen semigroup n an n dimensional exterior domain2005

    • Author(s)
      Y.Enomoto, Y.Shibata
    • Journal Title

      J.Math.Fluid Mech, 7

      Pages: 339-367

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the Stokes and Navier-Stokes equation in a perturbed half-space,2005

    • Author(s)
      T.Kubo, Y.Shibata
    • Journal Title

      Advances in Differential Equations, 10

      Pages: 695-720

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Uniform estimates in the velocity at infinity for stationary solutions to the Navier-Stokes exterior problem2005

    • Author(s)
      Y.Shibata, M.Yamazaki
    • Journal Title

      Japanes J.Math.(New Series) 31

      Pages: 225-279

    • NAID

      10020427306

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Asymptotic behaviour of solutions to the compressible Navier-Stokes equations on the half space2005

    • Author(s)
      Y.Kagei, T.Kobayashi
    • Journal Title

      Arch.Rational Mech.Anal. 177

      Pages: 231-330

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On some properties of solutions to the Stokes equation in the half-space and perturbed half-space2005

    • Author(s)
      Y.Shibata, T.Kubo
    • Journal Title

      Equation of Mathematical Physics in Quaderni in Matematica, Dept.Math.II.Univ.di Napoli 15

      Pages: 138-217

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-stokes equation2005

    • Author(s)
      Y.Enomoto, Y.Shibata
    • Journal Title

      J.math.Fluid mech 7

      Pages: 339-367

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the Stokes equation with Neumann boundary condition2005

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Banach Center Publications 70

      Pages: 239-250

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the Stokes and Navier-stokes flows in a perturbed half space2005

    • Author(s)
      T.Kubo, Y.Shibata
    • Journal Title

      Banach Center Publications 70

      Pages: 157-167

    • Related Report
      2005 Annual Research Report
  • [Journal Article] L_p-L_q maximal regularity and viscous incompressible flows with free surface2005

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Proc. Japan Acad. 81,Ser A

      Pages: 151-155

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Uniform estimates in the velocity at infinity for stationary solutions to the Navier-Stokes exterior problem2005

    • Author(s)
      Y.Shibata, M.Yamazaki
    • Journal Title

      Japanese J.Math.(New Series) 31

      Pages: 225-279

    • NAID

      10020427306

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Local energy decay of solutions to the Oseen Equation in the exterior domains2004

    • Author(s)
      Y.Enomoto, Y.Shibata
    • Journal Title

      Indiana J.Math. 53

      Pages: 1291-1330

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a resolvent estimate of a system of Laplace operators with perfect wall condition2004

    • Author(s)
      T.Akiyama, H.Kasai, Y.Shibata, M.Tsutsumi
    • Journal Title

      Funk.Ekvaj. 47

      Pages: 361-394

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a resolvent problem for the linearized System from the dynamical system describing the compressible viscous fluid motion2004

    • Author(s)
      Y.Shibata, K.Tanaka
    • Journal Title

      Math.Meth.Appl.Sci. 27

      Pages: 1579-1606

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On some properties of solutions to the Stokes equation in the half-space and perturbed half-space,2004

    • Author(s)
      T.Kubo, Y.Shibata
    • Journal Title

      Quad.Mat.(Dept.Math., Seconda Univ.Napoli,) 15

      Pages: 149-220

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Local energy decay of solutions to the Oseen equation in the exterior domains2004

    • Author(s)
      Yuko Enomoto, Yoshihiro Shibata
    • Journal Title

      Indiana Math.J. 53巻5号

      Pages: 1291-1330

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a resolvent estimate of a system of Laplace operators with perfect wall condition2004

    • Author(s)
      T.Akiyama, H.Kasai, Y.Shibata, M.Tsutsumi
    • Journal Title

      Fukcialaj Ekvacioj 47巻3号

      Pages: 361-394

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a resolvent problem for linearized system from the dynamical system describing the compressible viscous fluid motion2004

    • Author(s)
      Yoshihiro Shibata, Komei Tanaka
    • Journal Title

      Math.Meth.Appl.Sci. 27巻

      Pages: 1579-1606

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Wave front sets of the Riemann function of elastic interface problems2004

    • Author(s)
      Senjo Shimizu
    • Journal Title

      Math.Meth.Appl.Sci. 27巻5号

      Pages: 519-543

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The nonstationary Stokes and Navier-Stokes flows through an aperture2004

    • Author(s)
      T.Hishida
    • Journal Title

      Contributions to Current Challenges in Math.Fluid Mech.Adv.Fluid Mech 3巻

      Pages: 79-123

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Lq-theory of a singular"winding"integral operator arising from fluid dynamics2004

    • Author(s)
      R.Farwig, T.Hishida, D.Mueller
    • Journal Title

      Pacific J.Math 215巻

      Pages: 297-312

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On some resolvent estimate for the Stokes system with Neumann boundary condition2003

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Differnetial and Integral Equaations, 16

      Pages: 385-426

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a generalized resolvent estimate of the Stokes equation on an infinite layer,2003

    • Author(s)
      T.Abe, Y.Shibata
    • Journal Title

      J.Math.Soc.Japan 55

      Pages: 469-497

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the steady flow of compressible viscousfluid and its stability with respect to initial disturbance2003

    • Author(s)
      K.Tanaka, Y.Shibata
    • Journal Title

      J.Math.Soc.Japan 55

      Pages: 797-826

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a generalized resolvent estimate of the Stokesequation on an infinite layer, Part 22003

    • Author(s)
      T.Abe, Y.Shibata
    • Journal Title

      J.Math.Fluid Mech. 5

      Pages: 245-274

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a resolvent estimate of the interface problem for the Stokes system in a bounded domain2003

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      J.Diff.Eqns. 191

      Pages: 408-444

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the decay of solutions to the 2D Neumann exterior problem for the wave equation2003

    • Author(s)
      P.Secchi, Y.Shibata
    • Journal Title

      J.Diff.Eqns. 194

      Pages: 221-236

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the L_p-L_q maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Journal fur die reine und angewandte Mathematik

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a free boundary problem for the Navier-Stokes equations

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Differential and Integral Equations

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the L_p-L_q maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Journal fur die reine und angewandte Mathematik (Crulles Journal) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a free boundary problem for the Navier-Stokes equations

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Differntial and Integral equations, (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On a resolvent estimate for the Stokes equation with Robin boundary condition

    • Author(s)
      Y.Shibata, R.Shimada
    • Journal Title

      J.Math.Soc.Japan, (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the L_p-L_q maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Journal fur die reine and angewandte Mathematik (Crulles Journal) to appear

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On a free boundary problem for the Navier-Stokes equations

    • Author(s)
      Y.Shibata, S.Shimizu
    • Journal Title

      Differential and Integral Equations to appear

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On a resolvent estimate for the Stokes equation with Robin boundary condition

    • Author(s)
      Y.Shibata, R.Shimada
    • Journal Title

      Journal of Mathematical Society of Japan to appear

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A Stokes approximation of two dimensional exterior Oseen flow near the boundary

    • Author(s)
      M.Okamura, Y.Shibata, N.Yamaguchi
    • Journal Title

      Proceedings of MSJ-IRI 2005 "Asymptotic Analysis and Singularities" Advanced Studies in Pure Mathematics, to appear

    • Related Report
      2006 Annual Research Report
  • [Book] ルベーグ積分論2006

    • Author(s)
      柴田良弘
    • Total Pages
      377
    • Publisher
      内田老鶴圃
    • Related Report
      2005 Annual Research Report
  • [Publications] Y.Shibata, S.Shimizui: "On a resolvent estimate for the Stokes system with Neumann boundary condition"Differential and Integral Equations. 16. 385-426 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Shibata, S.Shimizu: "On a resolvent estimate of the interface problem for the Stokes system in a bounded domain"Journal of Differential Equations. 191. 408-444 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] P.Secchi, Y.Shibata: "On the decay of solutions to the 2D Neumann exterior problem for the wave equation"Journal of Differential Equations. 194. 221-236 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Abe, Y.Shibata: "On a resolvent estimate of the Stokes equation on an infinite layer"Journal of Mathematical Society of Japan. 55. 469-497 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Abe, Y.Shibata: "On a Resolvent Estimate of the Stokes Equation on an Infinite Layer Part 2"Journal of Mathematical Fluid Mechanics. 5. 245-275 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Shibata, K.Tanaka: "On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance"Journal of Mathematical Society of Japan. 55. 797-826 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Shibata: "Time-global solutions of nonlinear evolution equations and their stability"Ameriacan Mathematical Society, Translations. 211. 87-105 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2004-04-01   Modified: 2021-09-17  

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