Advertisement

Communications in Mathematical Physics

, Volume 129, Issue 3, pp 431–444 | Cite as

Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids

  • Sergio Albeverio
  • Ana-Bela Cruzeiro
Article

Abstract

We construct a family of probability spacesΩℱ,Pγ), γ<0 associated with the Euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow φt (time evolution) leavingPγ globally invariant. φt is obtained as the limit of Galerkin approximations associated with Euler equations.Pγ is also in invariant measure for a stochastic process associated with a Navier-Stokes equation with viscosity, γ, stochastically perturbed by a white noise force.

Keywords

Viscosity Neural Network Statistical Physic Complex System Time Evolution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AHK1] Albeverio, S., Høegh Krohn, R.: Stochastic flows with stationary distribution for two-dimensional inviscid fluids. Stoch. Proc. Appl.31, 1–31 (1989)CrossRefGoogle Scholar
  2. [AHK2] Albeverio, S., Høegh-Krohn, R.: Stochastic methods in quantum field theory and hydrodynamics. Phys. Repts.77, 193–214 (1981)CrossRefGoogle Scholar
  3. [AHKDeF] Albeverio, S., Ribeiro de Faria, M., Høegh Krohn, R.: Stationary measures for the periodic Euler flow in two dimensions. J. Stat. Phys.20, 585–595 (1979)CrossRefGoogle Scholar
  4. [AHKM] Albeverio, S., Høegh-Krohn, R., Merlini, D.: Some remarks on Euler flows, associated generalized random fields and Coulomb systems, pp. 216–244. In: Infinite dimensional analysis and stochastic processes. Albeverio, S. (ed.). London: Pitman 1985Google Scholar
  5. [BF1] Boldrighini, C., Frigio, S.: Equilibrium states for the two-dimensional incompressible Euler fluid. Atti Sem. Mat. Fis. Univ. ModenaXXVII, 106–125 (1978)Google Scholar
  6. [BF2] Boldrighini, C., Frigio, S.: Equilibrium states for a plane incompressible perfect fluid. Commun. Math. Phys.72, 55–76 (1980); Errata: ibid. Commun. Math. Phys.78, 303 (1980)CrossRefGoogle Scholar
  7. [BPP] Benfatto, G., Picco, P., Pulvirenti, M.: On the invariant measures for the two-dimensional Euler flow,CPT-CNRS Preprint, Marseille-Luminy (1986)Google Scholar
  8. [C1] Cruzeiro, A. B.: Solutions et measures invariants pour des équations d'évolution stochastiques du type Navier-Stokes. Expositiones Mathematicae7, 73–82 (1989)Google Scholar
  9. [C2] Cruzeiro, A. B.: to appear in Proc. 1987 Delphis-Conf. (1988)Google Scholar
  10. [C3] Cruzeiro, A. B.: Equations différentielles sur l'espace de Wiener et formules de Cameron-Martin non-linéaires. J. Funct. Anal.54, 206–227 (1983)CrossRefGoogle Scholar
  11. [CDG] Caprino, S., De Gregorio, S.: On the statistical solutions of the two-dimensional periodic Euler equations. Math. Methods Appl. Sci.7, 55–73 (1985)Google Scholar
  12. [DeF] Ribeiro de Faria, M.: Fluido de Euler bidimensional: construção de medidas estacionárias e fluxo estocástico. Diss., Universidade do Minho, Braga (Portugal), 1986Google Scholar
  13. [DP] Dürr, D., Pulvirenti, M.: On the vortex flow in bounded domain. Commun. Math. Phys.83, 265–273 (1983)Google Scholar
  14. [Du] Dubinskii, Y.: Weak convergence for nonlinear elliptic and parabolic equations. Mat. Sb.67, 609–642 (1965) (Russ.)Google Scholar
  15. [E] Ebin, D. G.: A concise presentation of the Euler equations of hydrodynamics. Commun. Partial Diff. Equations9, 539–559 (1984)Google Scholar
  16. [FT] Foias, C., Temam, R.: Self-similar universal homogeneous statistccal solutions of the Navier-Stokes equations. Commun. Math. Phys.90, 187–206 (1983)CrossRefGoogle Scholar
  17. [G] Glaz, H.: Statistical behavior and coherent structures in two-dimensional inviscid turbulence. Siam J. Appl. Math.41, 459–479 (1981)CrossRefGoogle Scholar
  18. [Ga] Gaveau, B.: Noyau de probabilités de transition de certains opérateurs d'Ornstein-Uhlenbeck dans l'espace de Hilbert. C.R. Acad. Sci. Paris, Ser. I,293, 469–472 (1981)Google Scholar
  19. [IW] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981Google Scholar
  20. [K] Kuo, H. H.: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics,463. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  21. [KK] Krée, M., Krée, P.: Continuité de la divergence dans les espaces de Sobolev relatifs à l'espace de Wiener. C.R. Acad. Sci. Paris, Ser. I, (1983)Google Scholar
  22. [KrMo] Kraichnan, R.H., Montgomery, D.: Two-dimensional turbulence. Rep. Progr. Phys.43, 547–619 (1980)CrossRefGoogle Scholar
  23. [Ma] Malliavin, P.: Implicit functions in finite corank on the Wiener space. In: Taniguchi Symposium 1982. Tokyo: Kinokuniya 1984Google Scholar
  24. [MP] Marchioro, C., Pulvirenti, M.: Vortex methods in two-dimensional fluid mechanics. Lecture Notes in Physics, vol203. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  25. [SV] Stroock, D. W., Varadhan, S.R.S.: Multidimensional diffusion processes. Grundlehren Mathem. vol.233. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  26. [Te] Temam, R.: Navier-Stockes equations; theory and numerical analysis. Amsterdam: North-Holland 1977Google Scholar
  27. [VKF] Vishik, M. I., Komechi, A. I., Fursikov, A. V.: Some mathematical problems of statistical hydrodynamics. Russ. Math. Surv.34, 149–234 (1979)Google Scholar
  28. [W] Welz, B.: Stochastische Gleichgewichtsverteilung eines 2-dimensionalen Superfluids. Diplomarbeit, Bochum (1986)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
  • Ana-Bela Cruzeiro
    • 3
  1. 1.Fakultät für MathematikRuhr-Universität-BochumBochumFederal Republic of Germany
  2. 2.BiBoS Research Centre and SFB 237Bochum-Essen-DüsseldorfFederal Republic of Germany
  3. 3.Centre de Matemática e Aplicações FundamentaisI.N.I.C.Lisboa CodexPortugal

Personalised recommendations