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Stability of nonlinear waves in viscous conservation system together with diffusion phenomena of solutions of damped wave equation

Research Project

Project/Area Number 16540206
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

NISHIHARA Kenji  Waseda University, Faculty of Political Science and Economics, Professor (60141876)

Co-Investigator(Kenkyū-buntansha) MATSUMURA Akitaka  Osaka University, Graduate School of Information Science and Technology, Professor (60115938)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,280,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥180,000)
Fiscal Year 2007: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordssystem of conservation law / nonlinear waves / stability / damped wave equation / diffusion phenomena / absorbing term / sourcing term / 粘性双曲型保存則 / 粘性衝撃波 / 希薄波
Research Abstract

The system of conservation laws has the shock wave, rarefaction wave and contact discontinuity as nonlinear waves. In real physics, it may become the system of viscous conservation laws by some viscous effect, which yields the viscous shock wave, rarefaction wave and viscous contact wave with diffusion wave. Our aim of this research is to observe the stability of the waves.
In our research the viscous effect is by the usual Newton viscosity or friction in porous media flow. The flow approaches to the solution of corresponding parabolic system by Darcy's law, which implies that the damped wave equation behaves as the corresponding diffusion equation as time tends to infinity, what we call the diffusion phenomena. The observation of this phenomena is another aim of this research. The stability of viscous contact wave in the viscous conservation laws with Newton's viscosity has been mainly developed by the investigator. The diffusion phenomena of solutions to the damped wave equation has been investigated by the head investigator, based on the fact that the solution of the Cauchy problem far the linear damped wave equation is decomposed to the sum of the wave part exponentially decaying and the diffusion part For the corresponding diffusion equation, rather precise results are obtained thanks to the smoothing effects and the maximum principle, but these key properties do not hold for the wave equation, and further studies are necessary.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (47 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (30 results) (of which Peer Reviewed: 11 results) Presentation (15 results) Book (2 results)

  • [Journal Article] Asymptotic behavior of solutions for the damped wave equation with slowly decaying data2008

    • Author(s)
      T.Narazaki, K.Nisbihara
    • Journal Title

      J.Math.Anal.Appl. 338

      Pages: 803-819

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of solutions for the damped wave equation with slowly decaying data2008

    • Author(s)
      T. Narazaki, K. Nishihara
    • Journal Title

      J. Math, Anal. Appl 338

      Pages: 803-819

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Asymptotic behavior of solutions for the damped wave equation with slowly decaying data2008

    • Author(s)
      T.Narazaki and K.Nishihara
    • Journal Title

      J.Math.Anal.Appl. 338

      Pages: 803-819

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large-time Behavior of Solutions to an Initial-boundary Value Problem on the Half Line for Scalar Viscous Conservation Law2007

    • Author(s)
      I.Hashimoto and A.Matsumura
    • Journal Title

      Met.Appl.Anal. 14

      Pages: 45-60

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Local Energy Decay for Wave Equations with Initial Data Decaying Slowly Near Infinity2006

    • Author(s)
      R.Ikehata, K.Nishihara
    • Journal Title

      Math.Sci.Appl. 22

      Pages: 265-275

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary 2005 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      K.Nishihara, H.Zhao
    • Journal Title

      J.Math.Anal.Appl. 313

      Pages: 598-610

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Global asymptoitcs of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      R.Ikehata, K.Nishihara, H.Zhao
    • Journal Title

      J.Differential Equations 226

      Pages: 1-29

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity2006

    • Author(s)
      K.Nishihara
    • Journal Title

      Z.angew.Math.Phys 57

      Pages: 604-614

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Global asymptotics for the damped wave equation with absorption in higer dimensional space2006

    • Author(s)
      K.Nishihara
    • Journal Title

      J.Math.Soc.Japan 58

      Pages: 805-836

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      K. Nishihara, H. Zhao
    • Journal Title

      J. Math, Anal. Appl 313

      Pages: 598-610

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Global asymptotics for the damped wave equation with absorption in Higher dimensional space2006

    • Author(s)
      K. Nishihara
    • Journal Title

      J. Math. Soc. Japan 58

      Pages: 805-836

    • NAID

      10018381129

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      K.Nishihara, H.Zhao
    • Journal Title

      J. Math. Anal. Appl. 313

      Pages: 598-610

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Global asymptoitcs of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      R.Ikehata, K.Nishihara, H.Zhao
    • Journal Title

      J. Differential Equations 226

      Pages: 1-29

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Global Asymptotics for the Damped Wabe Equation with Absorption in Higer dimensional space2006

    • Author(s)
      K.Nishihara
    • Journal Title

      J. Math. Soc. Japan 58

      Pages: 805-836

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity2006

    • Author(s)
      K.Nishihara
    • Journal Title

      Z. angev. Math. Phys. 57

      Pages: 604-614

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      K.Nishihara, H.Zhao
    • Journal Title

      J.Math.Anal.Appl.313,598-610(2006) 313

      Pages: 598-610

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations2006

    • Author(s)
      Feimin Huang, Akitaka Matsumura, Zhouping Xin
    • Journal Title

      Archive for Rational Mechanics and Analysis 179(1)

      Pages: 55-77

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Sharp lower bound for the lifespan of systems of semilinear wave wquations with multiple speeds2005

    • Author(s)
      Soichiro Katayama, Akitaka Matsumura
    • Journal Title

      J.Math.Kyoto University 45(2)

      Pages: 391-403

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Boundedness and partial survival of species in nonautonomous Lotka-Volterra systems2005

    • Author(s)
      Y.Muroya
    • Journal Title

      Nonlinear Analysis RWA 6

      Pages: 263-272

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations2004

    • Author(s)
      K.Nishihara, H.Zhao
    • Journal Title

      SIAM J.Math.Anal. 35

      Pages: 1561-1597

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Global stability of strong rarefaction waves of the Jin-Xin relaxation model for the $p$-system2004

    • Author(s)
      K.Nishihara, H.Zhao, Y.Zhao
    • Journal Title

      Comm.Partial Differential Equations 29

      Pages: 1607-1634

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary 2004 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and nonexistence of time-global solutions to damped wave equation on half-line2004

    • Author(s)
      K.Nishihara, H.Zhao
    • Journal Title

      Nonlinear Analysis 61

      Pages: 931-960

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations2004

    • Author(s)
      K.Nishihara, T.Yang, H.Zhao
    • Journal Title

      SIAM J.Math.Anal. 35

      Pages: 1561-1597

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Global stability for nonlinear difference equations with variable delay2004

    • Author(s)
      Y.Muroya, E.Ishiwata
    • Journal Title

      Proceedings o the Intern.Conf. on Nonlinear Analysis an Convex Analysis Tokyo 2003

      Pages: 347-358

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Persistence and global stability in Lotka-Volterra delay differential systems2004

    • Author(s)
      Y.Muroya
    • Journal Title

      Appl.Math.Letters 17

      Pages: 795-800

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the global attractivity for a logistic equation with piecewise constant arguments2004

    • Author(s)
      K.Uesugi, Y.Muroya, E.Ishiwata
    • Journal Title

      J.Math.Anal.Appl. 294

      Pages: 560-580

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Persistence and global stability in discrete models of pure-delay nonautonomous Lotka-Volterra type2004

    • Author(s)
      Y.Muroya
    • Journal Title

      J.Math.Anal.Appl. 293

      Pages: 446-461

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Erratum to "Persistence and global stability in discrete models of pure-delay onautonomous Lotka-Volterra type"[J.Math.Anal.Appl.293(2004)446-461]2004

    • Author(s)
      Y.Muroya
    • Journal Title

      J.Math.Anal.Appl. 300

      Pages: 260-263

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Global stability in discrete models of nonautonomous Lotka-Volterra type2004

    • Author(s)
      Y.Muroya
    • Journal Title

      Hokkaido Math.J. 33

      Pages: 115-126

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Existence and nonexistence of time-global solutions to damped wave equation on half-line

    • Author(s)
      K.Nishihara, H.Zhao
    • Journal Title

      Nonlinear Analysis (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Presentation] Asymptotic profile of solutions to a parabolic system of chemotaxis in one dimensional space2008

    • Author(s)
      K.Nishihara
    • Organizer
      東海大学発展方程式シンポジウム
    • Place of Presentation
      東海大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 単独粘性保存則に対する半直線上のある初期値境界値問題について2007

    • Author(s)
      橋本 伊都子, 松村 昭孝
    • Organizer
      日本数学会秋期総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2007 Annual Research Report
  • [Presentation] Behavior of solutions to the Cauchy problem for the damped wave equation2007

    • Author(s)
      K.Nishihara
    • Organizer
      6th ISAAC Congress
    • Place of Presentation
      Ankara,Turkey
    • Year and Date
      2007-08-19
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Behavior of solutions to the Cauchy problem for the damped wave equation2007

    • Author(s)
      K. Nishihara
    • Organizer
      ISAAC Congress
    • Place of Presentation
      Ankara, Turkey
    • Year and Date
      2007-08-19
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Behavior of solutions to the Cauchy problem for the damped wave equation2007

    • Author(s)
      Kenji Nishihara
    • Organizer
      6th ISAAC Congress
    • Place of Presentation
      Ankara
    • Year and Date
      2007-08-17
    • Related Report
      2007 Annual Research Report
  • [Presentation] Asymptotic behavior of solutions for the damped wave equation with slowly decaying data2007

    • Author(s)
      K.Nishihara
    • Organizer
      東海大学発展方程式シンポジウム
    • Place of Presentation
      東海大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 一次元粘性保存則系の解の長時間挙動粘性気体の方程式系を軸に.,I,II<Survey Lecture>2007

    • Author(s)
      松村 昭孝
    • Organizer
      研究集会「微分方程式の総合的研究」
    • Place of Presentation
      東京大学
    • Related Report
      2007 Annual Research Report
  • [Presentation] Large-time behavior of solutions for the damped wave equation2006

    • Author(s)
      K.Nishihara
    • Organizer
      AIMS 6th International Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Poitiers,France
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Behavior of solutions to the Cauchy problem for the damped wave equation with absorption2006

    • Author(s)
      K.Nishihara
    • Organizer
      International Conference on Nonlinear Evolutionary Partial Differential Equations
    • Place of Presentation
      Xining,China
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption2005

    • Author(s)
      K.Nishihara
    • Organizer
      東海大学発展方程式シンポジウム
    • Place of Presentation
      東海大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Large-time behavior of solutions for the damped wave equation2005

    • Author(s)
      K.Nishihara
    • Organizer
      京都大学数理解析研究所シンポジウム「流体と気体の数学解析」
    • Place of Presentation
      京都大学数理解析研究所
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of solutions to the Cauchy problem for the damped wave equation2005

    • Author(s)
      K.Nishihara
    • Organizer
      Conference "Self-similar solutions in nonlinear PDE's"
    • Place of Presentation
      Bedlewo,Poland
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 消散型波動方程式のCauchy問題の解の漸近挙動2005

    • Author(s)
      K.Nishihara
    • Organizer
      東海大学シンポジウム「偏微分方程式の諸問題」
    • Place of Presentation
      東海大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of solutions to the damped wave equation related to the heat equation2004

    • Author(s)
      K.Nishihara
    • Organizer
      International Conference on Nonlinear Evolutionary Partial Differential Equations
    • Place of Presentation
      Zhen-Jiang,China
    • Year and Date
      2004-05-16
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of solutions to the damped wave equation related to the heat equation2004

    • Author(s)
      K.Nishihara
    • Organizer
      WCNA-2004
    • Place of Presentation
      Orland,USA
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] 非線形微分方程式の大域解-圧縮性粘性流の数学解析-2004

    • Author(s)
      松村 昭孝, 西原 健二
    • Total Pages
      303
    • Publisher
      日本評論社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] 非線形微分方程式の大域解---圧縮性粘性流の数学解析---2004

    • Author(s)
      松村昭孝, 西原健二
    • Total Pages
      303
    • Publisher
      日本評論社
    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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