Asymptotics of Partial Differential Equations and the Renormalisation Group

  • Nigel Goldenfeld
  • Olivier Martin
  • Y. Oono
Part of the NATO ASI Series book series (NSSB, volume 284)


It is well-known that the asymptotics of partial differential equations (PDEs) may often be found from consideration of similarity solutions. In the examples usually encountered, the combinations of variables making up the similarity variables may be deduced using dimensional analysis; typically, the similarity variables are products of variables raised to rational fraction powers. It is not so widely appreciated, however, that there is a large class of problems where the similarity variables cannot be deduced from dimensional analysis. As Barenblatt has emphasized, such problems are neither rare nor pathological, but occur in many situations of physical interest, for example, in continuum mechanics.1


Renormalisation Group Anomalous Dimension Dimensional Analysis Perturbation Expansion Steady State Equation 
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  1. 1.
    G. I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics, (Consultants Bureau, New York, 1979).MATHCrossRefGoogle Scholar
  2. 2.
    N. Goldenfeld, O. Martin, Y. Oono and F. Liu, Phys. Rev. Lett. 64:1361 (1990);ADSCrossRefGoogle Scholar
  3. 2a.
    N. Goldenfeld, O. Martin, Y. Oono, J. Sci. Comp. 4:355 (1989).Google Scholar
  4. 3.
    D. J. Amit, Field Theory, The Renormalisation Group and Critical Phenomena, (McGraw-Hill, New York, 1978).Google Scholar
  5. 4.
    S.-K. Ma, Modern Theory of Critical Phenomena, (Benjamin/Cummings, Reading 1976).Google Scholar
  6. 5.
    See (e.g.) J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, (Clarendon, Oxford, 1989).Google Scholar
  7. 6.
    N. Goldenfeld: Proceedings of the Institute of Mathematics and its Applications, Workshop on the Evolution of Phase Boundaries, M. Gurtin and G. McFadden (eds.), (Springer-Verlag, to appear).Google Scholar
  8. 7.
    S. L. Kamenomostskaya, Dokl. Akad. Nauk SSSR 116:18 (1957).Google Scholar
  9. 8.
    M. Gell-Mann and F. E. Low, Phys. Rev. 95:1300 (1954).MathSciNetADSMATHCrossRefGoogle Scholar
  10. 9.
    The work of M. Berger and R. Kohn, Comm. of Pure and Applied Math. 41:841 (1988) is closely related to the renormalisation group.MathSciNetMATHCrossRefGoogle Scholar
  11. 10.
    K.G. Wilson, Phys. Rev.B 4:3174 (1971);ADSMATHCrossRefGoogle Scholar
  12. 2.
    K.G. Wilson, Phys. Rev.B 4:3184 (1971).ADSMATHCrossRefGoogle Scholar
  13. 11.
    D. Kessler, J. Koplik and H. Levine, Phys. Rev. A 31:1712 (1985).ADSCrossRefGoogle Scholar
  14. 12.
    E. Ben-Jacob, N. Goldenfeld, B. Kotliar and J. Langer, Phys. Rev. Lett. 53:2110 (1984).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Nigel Goldenfeld
    • 1
  • Olivier Martin
    • 2
  • Y. Oono
    • 1
  1. 1.Department of PhysicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Physics, City CollegeCUNYNew YorkUSA

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