Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids. (English) Zbl 0702.76041

Summary: We construct a family of probability spaces (\(\Omega\),\({\mathcal F},P_{\gamma})\), \(\gamma >0\) associated with the Euler equation for a two-dimensional inviscid incompressible fluid which carries a pointwise flow \(\phi_ t\) (time evolution) leaving \(P_{\gamma}\) globally invariant. \(\phi_ t\) is obtained as the limit of Galerkin approximations associated with Euler equations. \(P_{\gamma}\) is also an invariant measure for a stochastic process associated with a Navier- Stokes equation with viscosity \(\gamma\), stochastically perturbed by a white noise force.


76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
Full Text: DOI


[1] [AHK1] Albeverio, S., Høegh Krohn, R.: Stochastic flows with stationary distribution for two-dimensional inviscid fluids. Stoch. Proc. Appl.31, 1–31 (1989) · Zbl 0676.76033
[2] [AHK2] Albeverio, S., Høegh-Krohn, R.: Stochastic methods in quantum field theory and hydrodynamics. Phys. Repts.77, 193–214 (1981)
[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)
[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 1985
[5] [BF1] Boldrighini, C., Frigio, S.: Equilibrium states for the two-dimensional incompressible Euler fluid. Atti Sem. Mat. Fis. Univ. ModenaXXVII, 106–125 (1978) · Zbl 0437.76021
[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) · Zbl 0453.76019
[7] [BPP] Benfatto, G., Picco, P., Pulvirenti, M.: On the invariant measures for the two-dimensional Euler flow,CPT-CNRS Preprint, Marseille-Luminy (1986)
[8] [C1] Cruzeiro, A. B.: Solutions et measures invariants pour des équations d’évolution stochastiques du type Navier-Stokes. Expositiones Mathematicae7, 73–82 (1989) · Zbl 0665.60066
[9] [C2] Cruzeiro, A. B.: to appear in Proc. 1987 Delphis-Conf. (1988)
[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) · Zbl 0524.47028
[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) · Zbl 0578.76020
[12] [DeF] Ribeiro de Faria, M.: Fluido de Euler bidimensional: construção de medidas estacionárias e fluxo estocástico. Diss., Universidade do Minho, Braga (Portugal), 1986
[13] [DP] Dürr, D., Pulvirenti, M.: On the vortex flow in bounded domain. Commun. Math. Phys.83, 265–273 (1983)
[14] [Du] Dubinskii, Y.: Weak convergence for nonlinear elliptic and parabolic equations. Mat. Sb.67, 609–642 (1965) (Russ.)
[15] [E] Ebin, D. G.: A concise presentation of the Euler equations of hydrodynamics. Commun. Partial Diff. Equations9, 539–559 (1984) · Zbl 0548.76005
[16] [FT] Foias, C., Temam, R.: Self-similar universal homogeneous statistccal solutions of the Navier-Stokes equations. Commun. Math. Phys.90, 187–206 (1983) · Zbl 0532.60058
[17] [G] Glaz, H.: Statistical behavior and coherent structures in two-dimensional inviscid turbulence. Siam J. Appl. Math.41, 459–479 (1981) · Zbl 0479.76059
[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) · Zbl 0484.35077
[19] [IW] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981
[20] [K] Kuo, H. H.: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics,463. Berlin, Heidelberg, New York: Springer 1979
[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) · Zbl 0531.46034
[22] [KrMo] Kraichnan, R.H., Montgomery, D.: Two-dimensional turbulence. Rep. Progr. Phys.43, 547–619 (1980)
[23] [Ma] Malliavin, P.: Implicit functions in finite corank on the Wiener space. In: Taniguchi Symposium 1982. Tokyo: Kinokuniya 1984
[24] [MP] Marchioro, C., Pulvirenti, M.: Vortex methods in two-dimensional fluid mechanics. Lecture Notes in Physics, vol203. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0545.76027
[25] [SV] Stroock, D. W., Varadhan, S.R.S.: Multidimensional diffusion processes. Grundlehren Mathem. vol.233. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0426.60069
[26] [Te] Temam, R.: Navier-Stockes equations; theory and numerical analysis. Amsterdam: North-Holland 1977
[27] [VKF] Vishik, M. I., Komechi, A. I., Fursikov, A. V.: Some mathematical problems of statistical hydrodynamics. Russ. Math. Surv.34, 149–234 (1979) · Zbl 0503.76045
[28] [W] Welz, B.: Stochastische Gleichgewichtsverteilung eines 2-dimensionalen Superfluids. Diplomarbeit, Bochum (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.