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A structure of solutions of initial-boundary value problems for parabolic-hyperbolic equations and applications to numerical analises

Research Project

Project/Area Number 16540174
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KOBAYASI Kazuo  Waseda University, School of Education, Professor, 教育・総合科学学術院・教育学部, 教授 (30139589)

Co-Investigator(Kenkyū-buntansha) 高木 悟  早稲田大学, 教育・総合科学学術院, 助手 (50367017)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Keywordsdegenerate parabolic equation / initial-boundary value problem / entropy solution / renormalized entropy solution / comparison theorem / kinetic method / Kinetic法 / 退化放物型-双曲型 / 初期値-境界値問題 / Renormalized Solution / Entropy Solution / Dissipative Solution / Kinetic Formulation
Research Abstract

This research is mainly concerned with entropy solutions for scalar conservation laws as well as nonlinear degenerate parabolic equations. The main results which we obtained are the followings : 1. We introduce a new notion of renormalized dissipative solutions for scalar conservation laws with locally Lipschitz continuous flux-functions and prove that this solution is equivalent to renormalized entropy solutions which are introduced by Benilan et al. in the study of (unbounded) L1 solutions. We also construct a renormalized dissipative solution for contractive relaxation systems in merely an L1 setting. 2. We study two types unbounded weak solutions of the Cauchy problem for scalar conservation laws, that is a renormalized entropy solution and a kinetic solution. It is proved that if u is a kinetic solution, then it is indeed a renormalized entropy solution. Conversely, we prove that if u is a renormalized entropy solution which satisfies a certain additional condition, then it becomes a kinetic solution. 3. We study the comparison principle for nonlinear degenerate parabolic equations with initial and non-homogeneous boundary conditions. We prove a comparison theorem for any (bounded) entropy sub-and super-solution. The L1 contractivity and therefore uniqueness of (unbounded) entropy solutions has been obtained so far, but a comparison result is new. The method used there is the so called doubling variable technique due to Kruzkov. Our method is based upon the kinetic formulation and the kinetic technique. By developing the kinetic technique for degenerate parabolic equations with boundary conditions, we obtain a comparison property. 4. We extend the above result stated in the item 3 to the case of unbounded entropy solutions. We consider the problem in an L1 framework and prove a comparison theorem and existence theorem for renormalized entropy solutions.

Report

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

    (20 results)

All 2007 2006 2005 2004 Other

All Journal Article (20 results)

  • [Journal Article] Remarks on BV estimates for vanishing viscosity approximations to hyperbolic systems2007

    • Author(s)
      Kazuo Kobayasi, Hiroki Ohwa
    • Journal Title

      早稲田大学教育学部学術研究(数学偏) 55

      Pages: 1-14

    • NAID

      120001293392

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Remarks on BV estimates for vanishing viscosity approximations to hyperbolic systems2007

    • Author(s)
      K.Kobayasi, H.Ohwa
    • Journal Title

      Gakujutu Kenkyu-Mathematics, School of Education, Waseda University 55

      Pages: 1-14

    • NAID

      120001293392

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Remarks on BV estimates for vanishing viscosity approximations to hyperbolic systems2007

    • Author(s)
      Kazuo Kobayasi, Hiroki Ohwa
    • Journal Title

      早稲田大学教育学部学術研究-数学編- 55

      Pages: 1-14

    • NAID

      120001293392

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions2006

    • Author(s)
      Kazuo Kobayasi
    • Journal Title

      J. Differential Equations 230

      Pages: 682-701

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] A kinetic approach to a comparison theorem for degenerate parabolic equations2006

    • Author(s)
      Kazuo Kobayasi
    • Journal Title

      京都大学数理解析研究所講究録 1475

      Pages: 155-166

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The relationship between kinetic solutions and renormalized Entropy solutions of scalar conservation laws2006

    • Author(s)
      Satomi Ishikawa, Kazuo Kobayasi
    • Journal Title

      Proceedings of the conference on Differential & Difference equations and applications

      Pages: 433-440

    • 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 existence of renormalized dissipative solutions via Relaxation for conservation laws2006

    • Author(s)
      Kazuo Kobayasi, Satoru Takagi
    • Journal Title

      Hyperbolic Problem : Theory, Numerics and Applications 2

      Pages: 101-108

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A kinetic approach to comparison properties for degenerate parabolic- hyperbolic equations with boundary conditions2006

    • Author(s)
      K.Kobayasi
    • Journal Title

      J. Differential quations 230

      Pages: 682-701

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A kinetic approach to a comparison theorem for degenerate parabolic equations2006

    • Author(s)
      K.Kobayasi
    • Journal Title

      Kokyuroku, RIMS, Kyoto University 1475

      Pages: 155-166

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The relationship between kinetic solutions and renormalized entropy solutions of scalar conservation laws2006

    • Author(s)
      S.Ishikawa, K.Kobayasi
    • Journal Title

      Proceedings of the conference on Differential & Difference equations and Applications (Hindaui Corporation)

      Pages: 433-440

    • 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 existence of renormalized dissipative solutions via relaxation for conservation laws2006

    • Author(s)
      K.Kobayasi, S.Takagi
    • Journal Title

      Hyperbolic Problem : Theory, Numerics and Applications-II

      Pages: 101-108

    • 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 existence of renormalized dissipative solutions via relaxation for conservatoin laws2006

    • Author(s)
      Kazuo Kobayashi, Satoru Takagi
    • Journal Title

      Hyperbolic Problem : Theory, Numerics and Applications 2

      Pages: 101-108

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The relationship between kinetic solutions and renormalized entropy solutions of scalar conservation laws2006

    • Author(s)
      Satomi Ishikawa, Kazuo Kobayasi
    • Journal Title

      Proceedings of the conference on Differential & Difference equations and Applications

      Pages: 433-440

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A kinetic approach to a comparison theorem for degenerate parebolic equations2006

    • Author(s)
      Kazuo Kobayasi
    • Journal Title

      京都大学数理解析研究所講究録 1475

      Pages: 155-166

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the existence of renormalized dissipative solutions via relaxation for conservation laws2006

    • Author(s)
      Kazuo Kobayasi, Satoru Takagi
    • Journal Title

      Hyperbolic Problems : Theory, Numercs and Applications

      Pages: 101-108

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An equivalent definition of renormalized entropy solutions for scalar conservation laws2005

    • Author(s)
      Kazuo Kobayasi, Satoru Takagi
    • Journal Title

      Differential and Integral Equations 18

      Pages: 19-33

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] An equivalent definition of renormalized entropy solutions for scalar conservation laws2005

    • Author(s)
      K.Kobayasi, S.Takagi
    • Journal Title

      Differential and Integral Equations 18

      Pages: 19-33

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Renormalized dissipative solutions of second order degenerate parabolic balance laws2004

    • Author(s)
      Satoru Takagi
    • Journal Title

      早稲田大学教育学部学術研究-数学編- 53

      Pages: 51-65

    • NAID

      110004617129

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the existence of renormalized dissipative solutions via relaxation for conservation laws2004

    • Author(s)
      Kazuo Kobayasi, Satoru Takagi
    • Journal Title

      the Proceedings of HYP2004 (to appear)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A kinetic approach to a comparison theorem for degenerate parabolic equations

    • Author(s)
      Kazuo Kobayasi
    • Journal Title

      京都大学数理解析研究所講究録

    • Related Report
      2005 Annual Research Report

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

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