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Study of evolution equations in the space of BV functions

Research Project

Project/Area Number 23540239
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Global analysis
Research InstitutionShizuoka University

Principal Investigator

KIKUCHI Koji  静岡大学, 工学研究科, 教授 (50195202)

Co-Investigator(Kenkyū-buntansha) SHIMIZU Senjo  静岡大学, 理学研究科, 教授 (50273165)
HOSHIGA Akira  静岡大学, 工学研究科, 准教授 (50261400)
ADACHI Shinji  静岡大学, 工学研究科, 准教授 (40339685)
NAKAJIMA Toru  静岡大学, 工学研究科, 准教授 (50362182)
星賀 彰  静岡大学, 工学研究科, 准教授 (60261400)
Project Period (FY) 2011-04-28 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords函数方程式論 / 函数解析学
Outline of Final Research Achievements

In this research we obtained that for a quasilinear hyperbolic equation, if the initial data is sufficiently small, then a solution behaves like a solution to the linearlized equation, and that for the equation of motion of vibrating membrane with a kind of strong viscosity term, if the initial data is slightly regular (but twice differentability is suficient) , there is a unique solution in the space of BV functions. But this strong viscosity term does not have variational structure. We hope to obtain the same results for the equation of motion of vibrating membrane with a strong viscosity term which has variational structure. In the period of our research we only develope our analysis in Sobolev spaces, and it is not extended to the theory in the space of BV functions.

Report

(5 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • 2011 Research-status Report

Research Products

(27 results)

All 2015 2014 2013 2012 2011

All Journal Article (16 results) (of which Peer Reviewed: 14 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (11 results)

  • [Journal Article] On local Lp-Lq well-posedness of incompressible two-phase flows with phase transitions: the case of non equal densities2015

    • Author(s)
      Senjo Shimizu, Shintaro Yagi
    • Journal Title

      Differential Integral Equations

      Volume: 28 Pages: 29-58

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Constructing a solution to an evolution equation having a strong viscosity term of p-growth2014

    • Author(s)
      Koji Kikuchi
    • Journal Title

      Proceedings of the International Conference: Functional Analysis and Applications

      Volume: 1 Pages: 55-71

    • Related Report
      2014 Annual Research Report
  • [Journal Article] Qualitative behaviour of incompressible two-phase flows with phase transitions: the case of non-equal densities2014

    • Author(s)
      Jan Pruess, Senjo Shimizu, Mathias Wilke
    • Journal Title

      Comm. P. D. E.

      Volume: 39 Pages: 1236-1283

    • DOI

      10.1080/03605302.2013.821131

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Asymptotic Behavior of Positive Solutions for a Class of Quasilinear Elliptic Equations in <b><i>R</i></b><sup>2</sup>2014

    • Author(s)
      Shinji Adachi, Masataka Shibata, Tatsuya Watanabe
    • Journal Title

      FE

      Volume: 57 Issue: 2 Pages: 297-317

    • DOI

      10.1619/fesi.57.297

    • NAID

      130004678000

    • ISSN
      0532-8721
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Uniqueness and non-degeneracy of positive radial solutions for quasilinear elliptic equations with exponential nonlinearity2014

    • Author(s)
      Shinji Adachi, Tatsuya Watanabe
    • Journal Title

      Nonlinear Analysis: Theory, Methods & Applications

      Volume: 108 Pages: 275-290

    • DOI

      10.1016/j.na.2014.05.020

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with H1-supercritical nonlinearities2014

    • Author(s)
      Shinji Adachi, Masataka Shibata, Tatsuya Watanabe
    • Journal Title

      J. Differential Equations

      Volume: 256 Pages: 1492-1514

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities2014

    • Author(s)
      Shinji Adachi, Masataka Shibata, Tatsuya Watanabe
    • Journal Title

      Commun. Pure Appl. Anal.

      Volume: 13 Pages: 97-118

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Existence and uniqueness of a solution in the space of BV functions to the equation of a vibrating membrane with a "viscosity" term2013

    • Author(s)
      Koji Kikuchi
    • Journal Title

      J. Calculus of Variations

      Volume: 2013

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Linear approximation of a system of quasilinear hyperbolic equations having linear growth energy functional2013

    • Author(s)
      Koji Kikuchi
    • Journal Title

      Proceedings of Seminar on Partial Differential Equations in Osaka 2012

      Volume: 1 Pages: 141-150

    • Related Report
      2012 Research-status Report
  • [Journal Article] On well-posedness of incompressible two-phase flows with phase transitions: the case of non-equal densities2012

    • Author(s)
      Jan Pruess, Senjo Shimizu
    • Journal Title

      J. Evol. Equ.

      Volume: 12 Pages: 917-941

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the maximal L p -L q regularity of the Stokes problem with first order boundary condition; model problems2012

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      J. Math. Soc. Japan

      Volume: 64 Pages: 561-626

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymptotic properties of ground states of quasilinear Schrödinger equations with H 1 -subcritical exponent2012

    • Author(s)
      Shinji Adachi, Tatsuya Watanabe
    • Journal Title

      Adv. Nonlinear Stud.

      Volume: 12 Pages: 255-279

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Uniqueness of the ground state solutions of quasilinear Schrödinger equations2012

    • Author(s)
      Shinji Adachi, Tatsuya Watanabe
    • Journal Title

      Nonlinear Anal.

      Volume: 75 Pages: 819-833

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Loss of control of motions from initial data for pending capillary liquid2011

    • Author(s)
      Umberto Massari, Mariarosaria Padula, Senjo Shimizu
    • Journal Title

      Quart. Appl. Math.

      Volume: 69 Pages: 569-601

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Maximal Lp-Lq regularity for the two-phase Stokes equations; model problems2011

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      J. Differential Equations

      Volume: 251 Pages: 373-419

    • NAID

      120003115040

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension2011

    • Author(s)
      Yoshihiro Shibata, Senjo Shimizu
    • Journal Title

      Appl. Anal.

      Volume: 90 Pages: 201-214

    • NAID

      120003962945

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Presentation] 非線形の強粘性項を持つ発展方程式について2015

    • Author(s)
      菊地光嗣
    • Organizer
      松山解析セミナー 2015
    • Place of Presentation
      愛媛大学(愛媛県松山市)
    • Year and Date
      2015-02-06 – 2015-02-07
    • Related Report
      2014 Annual Research Report
  • [Presentation] 非線形の強粘性項を持つ発展方程式2014

    • Author(s)
      菊地光嗣
    • Organizer
      The Ninth Conference on Probability and PDE
    • Place of Presentation
      津田塾大学(東京都小平市)
    • Year and Date
      2014-11-13 – 2014-11-14
    • Related Report
      2014 Annual Research Report
  • [Presentation] Uniqueness and non-degeneracy of positive radial solutions for quasilinear elliptic equations with exponential nonlinearity2014

    • Author(s)
      足達慎二,渡辺達也
    • Organizer
      日本数学会年会
    • Place of Presentation
      学習院大学(東京都)
    • Related Report
      2013 Research-status Report
  • [Presentation] 相転移を伴う誘拐領域内非圧縮性2相流の解の安定性 - 表面張力が変数の場合 -2013

    • Author(s)
      J. Pruess, 清水扇丈, G. Simmonett, M. Wilke
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      愛媛大学(愛媛県)
    • Related Report
      2013 Research-status Report
  • [Presentation] 1階対称双曲系の散乱理論について2013

    • Author(s)
      菊地光嗣
    • Organizer
      スペクトル・散乱松山シンポジウム
    • Place of Presentation
      愛媛大学(愛媛県)
    • Related Report
      2012 Research-status Report
  • [Presentation] A Remark on a evolution equation having a strong viscosity term2013

    • Author(s)
      Koji Kikuchi
    • Organizer
      Functional Analysis and Applications - Evolution Equations and Control Theory -
    • Place of Presentation
      神戸大学(兵庫県)
    • Related Report
      2012 Research-status Report
  • [Presentation] An analysis in the space of BV functions for the equation of motion of a vibrating membrane with a 'viscosity' term2012

    • Author(s)
      Koji Kikuchi
    • Organizer
      AIMS2012
    • Place of Presentation
      The Hyatt Regency Grand Cypress in Orlando(米国)
    • Related Report
      2012 Research-status Report
  • [Presentation] Linear approximation of a system of quasilinear hyperbolic equations having linear growth energy functional2012

    • Author(s)
      Koji Kikuchi
    • Organizer
      Seminar on Partial Differential Equations in Osaka, 2012
    • Place of Presentation
      大阪大学工学部(大阪府)
    • Related Report
      2012 Research-status Report
  • [Presentation] An analysis in the space of BV functions for an evolution equation having a strong viscosity term2012

    • Author(s)
      Koji Kikuchi
    • Organizer
      Taiwan-Japan Joint Conference on PDE and Analysis
    • Place of Presentation
      国立台湾大学(台湾)
    • Related Report
      2012 Research-status Report
  • [Presentation] An analysis in the space of BV functions for the equation of motion of a vibrating membrane with a ‘viscosity’ term2011

    • Author(s)
      菊地光嗣
    • Organizer
      第16回広島応用解析セミナー
    • Place of Presentation
      広島大学工学部(広島県)
    • Related Report
      2011 Research-status Report
  • [Presentation] エネルギーが一次増大度となる双曲型方程式系について2011

    • Author(s)
      菊地光嗣
    • Organizer
      第9回浜松編微分方程式研究集会
    • Place of Presentation
      静岡大学工学部(静岡県)
    • Related Report
      2011 Research-status Report

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Published: 2011-08-04   Modified: 2019-07-29  

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