• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2001 Fiscal Year Final Research Report Summary

Problems of Hydrodynamic Limits and Related Subjects

Research Project

Project/Area Number 11440026
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTokyo Institute of Technology

Principal Investigator

UCHIYAMA Kohei  Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (00117566)

Co-Investigator(Kenkyū-buntansha) SHIGA Hirosige  Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (10154189)
MURATA Minoru  Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (50087079)
FUNAKI Tadahisa  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (60112174)
SHIRAI Tomoyuki  Graduate School of Science and Engineering, Tokyo Institute of Technology, Assistant, 大学院・理工学研究科, 助手 (70302932)
MORITA Takehiko  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00192782)
Project Period (FY) 1999 – 2001
Keywordshydrodynamic limit / local equilibrium / scaling limit / relative entropy / large deviation
Research Abstract

1. The hydrodynamic limit for a system of multi-dimensional Brownian particles which interact one another through a two-body potential having a finite range and satisfying the super-stability condition is derived under the hypothesis that the third moments of the empirical densities are uniformly bounded. For the derivation we also show a kind of virial theorem in the classical theory of Gibbs statistical mechanics: the pressure is represented as a limit of normalized sums of the virials in the law of large numbers in every space dimension [K. Uchiyama, Pressure in classical statistical mechanics and interacting Brownian particles in multi-dimensions, Annales Henri Poincare (J. Theor. Math. Phys.) 1,1159-1202 (2000)]. Although such a theorem has already been proved if the uniqueness for Gibbs states holds (as in the cases of one-dimension or of high temperature ), this is for the first time of the proof without assuming the uniqueness.
2. A class of one-dimensional evolution equations which are nonlocal and derived in the scaling limits from systems of interacting particles obeying the classical law of motion is studied. As fundamental theorems there are established the existence and the uniqueness of solutions, the maximum principle, a comparison theorem for integrated solutions, and the convergences to the 'Barenblatt' solutions, etc. In the particular case when the interacting force in the particle system is given by a logarithmic potential, quite fine properties of the solutions are obtained. [K.Uchiyama, Behavior of solutions to the initial value problem for a class of integro-differential equations, Nonlinear Analysis (2001).] These results if combined with former results on the scaling limits give a clear macroscopic view about the behavior of the underlying microscopic model of particles when the interaction potential has a long range.

  • Research Products

    (23 results)

All Other

All Publications (23 results)

  • [Publications] K.Uchiyama: "Uniqueness of solutions to the initial value problem for an integro-differential equation"Diff. It. Eqs.. 13. 401-422 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Uchiyama: "A non-linear evolution equation driven by a logarithmic potential"Bull. London Math. Soc.. 32(no.156). 353-363 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Uchiyama: "Pressure in classical statistical mechanics and interacting Brownian particles in multi-dimensions"Annales Henri Poincare (J. Theor. Math. Phys.). 1. 1159-1202 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Uchiyama: "Behavior of solutions to the initial value problem for a class of integro-differential equations"Nonlinear Analysis. (印刷中). (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Funaki: "Free boundary problem from stochastic lattice gas model"Ann. Inst. Henri Poincare. 35. 573-603 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Inoue, Y.Nomura: "Some refinements of Wigner's semi-circle law for Gaussian random matrices using superanalysis"Asymptotic Analysis. 23. 329-375 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Funaki, T.Nishikawa: "Large deviations for the Ginzburg-Landau $Ynabla Yphi$ interface model"Probab. Theory Relat. Fields. 120. 535-568 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Shiga: "On holomorphic families of rational maps : Finiteness, Rigidity and Stability"Kodai Math. J. 24. 48-65 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Murata, K.Ishige: "Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains"Ann. Scuola Normale Sup. Pisa. 30. 171-223 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Shirai: "Asymptotic behavior of the transition probability of a simple random walk on a line graph"J. Math. Soc. Japan. 52. 99-108 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Shirai: "The spectrum of infinite regular line graphs"Trans. of Amer. Math. Soc.. 352. 115-132 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 舟木直久, 内山耕平: "シュプリンガー・フェアラーク東京"ミクロからマクロヘ,1.界面モデルの数理(印刷中). (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K. Uchiyama: "Uniqueness of solutions to the initial value problem for an integro-differential equation"Diff. Int. Eqs.. 13. 401-422 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Uchiyama: "A non-linear evolution equation driven by a logarithmic potential"Bull. London Math. Soc.. 32 (no.156). 353-363 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Uchiyama: "Pressure in classical statistical mechanics and interacting Brownian particles in multi-dimensions"Annales Henri Poincare (J. Theor. Math. Phys.). 1. 1159-1202 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Uchiyama: "Behavior of solutions to the initial value problem for a class of integro-differential equations"Nonlinear Analysis. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Funaki: "Free boundary problem from stochastic lattice gas model"Ann. Inst. Henri Poincare. 35. 573-603 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Inoue and Y. Nomura: "Some refinements of Wigner's semi-circle law for Gaussian random matrices using Superanalysisy"Asymptotic Analysis. 23. 329-375 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Funaki and T. Nishikawa: "Large deviations for the Ginzburg-Landau **nabla *phi* interface model"Probab. Theory Relat. Fields. 120. 535-568 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H. Shiga: "On holomorphic families of rational maps : Finiteness, Rigidity and Stability"Kodai Math. J.. 21. 48-65 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Murata and K. Ishige: "Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains"Ann. Scuola Normale Sup. Pisa. 30. 171-223 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Shirai: "Asymptotic behavior of the transition probability of a simple random walk on a line graph"J. Math. Soc. Japan. 52. 99-108 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Shirai: "The spectrum of infinite regular line graphs"Trans. of Amer. Math. Soc.. 352. 115-132 (2000)

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

URL: 

Published: 2003-09-17  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi