Stochastic Weak Attractor for a Dissipative Euler Equation

Hakima Bessaih (University Dini, Pisa)


In this paper a nonautonomous dynamical system is considered, a stochastic one that is obtained from the dissipative Euler equation subject to a stochastic perturbation, an additive noise. Absorbing sets have been defined as sets that depend on time and attracts fromĀ  $-\infty$. A stochastic weak attractor is constructed in phase space with respect to two metrics and is compact in the lower one.

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Pages: 1-16

Publication Date: November 29, 1999

DOI: 10.1214/EJP.v5-59


  1. C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimensione deux,  J. Math.  Anal. Appl, 40, 1972, 769-780. Math. Review 48 #11813
  2. H. Bessaih and F. Flandoli,  2-D Euler equations pertubed by noise, Nonlinear Diff. Eq .Appl, Vol 6, Issue 1, 1999, 35-54. Math. Review CMP 1 674 779
  3. H. Bessaih and F. Flandoli,  Weak attractor for a dissipative Euler equation, Submitted. Math. Review number not available.
  4. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Theo. Relat. Fields, 100, 1994,  365-393. Math. Review 95k:58092
  5. H. Crauel, A. Debusshe and F. Flandoli, Random attractors, J. Dyn. Diff. Eq, 1995. Math. Review 98c:60066
  6. C. Castaing and M. Valadier,  Convex analysis and measurable multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin, 1977. Math. Review 57 #7169
  7. V. Chepyshov and  M.I. Vishik,  Nonautonomous evolution equations and their attractors,  Russian J. Math. Phys, 1, 1993, 165-190. Math. Review 93m:34094
  8. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. Math. Review 95g:60073
  9. F. Flandoli and B. Schmalfuss, Weak solutions and attractors for the 3-dimensional Navier-Stokes equations with nonregular force, J. Dyn. Diff. Eq., 11, 1999, 355-398. Math. Review 1 695 250
  10. G. Gallavotti,  Ipotesi per uua introduzione alla Meccanica Dei  Fluidi, Quaderni del Consiglio Nazionale delle Ricerche, Gruppo Nazionale di Fisica matematica , no 52, 1996. Math. Review number not available
  11. J. L. Lions, Équations Differentielles Opérationnelles et problèmes aux limites, Springer-Verlag, Berlin, 1961. Math. Review 27 #3935
  12. P.L. Lions,  Mathematical Topics in Fluid Mechanics Vol. 1 Incompressible Models, Oxford Sci. Publ, Oxford, 1996. Math. Review 98b:76001
  13. G. Sell, Global attractors for the 3D Navier-Stokes Equations, J. Dyn. Diff. Eq, 8, 1996. Math. Review 98e:35127
  14. R. Temam, Navier-Stokes Equations, North-Holland, 1984. Math. Review 86m:76003
  15. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol 68, Springer-Verlag, New York, 1988. Math. Review 89m:58056
  16. M. J. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht, 1980. Math. Review 83e:35098

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