Stochastic two dimensional Euler equations

Annals of Probability

Volumn 29, Issue 4, 2001, Pages 1796-1832

  Brzeźniak, Zdzisław ^a   Peszat, Szymon ^b  

^a UNIVERSITY OF HULL   (United Kingdom)

^b INSTITUTE OF MATHEMATICS   (Poland)

Author keywords

Martingale problems on banach spaces; Stochastic euler equations; Stochastic hydromechanics; Stochastic navier stokes equations

Indexed keywords

EID: 0035470881     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1015345773     Document Type: Article

References (47)

1. ADAMS, R. A. (1975). Sobolev Spaces. Academic Press, New York.
2. AGMON, S., DOUGLIS, A. and NIRENBERG, L. (1959). Estimates near boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. 12 623-727.
3. BARDOS, C. (1972). Existence et unicité de la solution de l'equation d'Euler en dimension deux. J. Math. Anal. Appl. 40 769-790.
4. BAXENDALE, P. (1976). Gaussian measures on function spaces. Amer. J. Math. 98 891-952.
5. BERGH, J. and LÖFSTRÖM, J. (1976). Interpolation Spaces. Springer, Berlin.
6. BESSAIH, H. (1999). Stochastic PDEs of Euler type. Ph.D. thesis, Scuola Normale Superiore, Pisa.
7. BESSAIH, H. and FLANDOLI, F. (1999). 2-D Euler equations perturbed by noise. Nonlinear Differential Equations and Applications 6 35-54.
8. BESSAIH, H. (1999). Martingale solutions for stochastic Euler equations. Stochastic Anal. Appl. 17 713-725.
9. BRZEŹNIAK, Z. (1995). Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4 1-45.
10. BRZEŹNIAK, Z. (1997). On stochastic convolutions in Banach spaces and applications. Stochastics Stochastics Rep. 61 245-295.
11. BRZEŹNIAK, Z. and CARROLL, A. (2000). Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces. Preprint.
12. BRZEŹNIAK, Z. and GA̧TAREK, D. (1999). Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces. Stochastic Processes Appl. 87 187-226.
13. BRZEŹNIAK, Z. and PESZAT, S. (1999). Space-time continuous solutions to SPDEs driven by a homogeneous Wiener process. Studia Math. 137 261-299.
14. BRZEŹNIAK, Z. and PESZAT, S. (2000). Strong local and global solutions to stochastic Navier-Stokes equations. In Infinite-Dimensional Stochastic Analysis 85-98. Royal Netherlands Academy of Arts and Sciences, Amsterdam.
15. CAPIŃSKI, M. and CUTLAND, N. J. (1999). Stochastic Euler equations on the torus. Ann. Appl. Probab. 9 688-705.
16. CAPIŃSKI, M. and GA̧TAREK, D. (1994). Stochastic equations in Hubert space with applications to 3-D stochastic Navier-Stokes equations. J. Funct. Anal. 126 25-35.
17. CAPIŃSKI, M. and PESZAT, S. (2001). On the existence of a solution to stochastic Navier-Stokes equations. Nonlinear Anal. 44 141-177.
18. CARROLL, A. (1999). The stochastic nonlinear heat equation. Ph.D. thesis, Univ. Hull.
19. DA PRATO, G. and ZABCZYK. J (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
20. DA PRATO, G. and ZABCZYK, J. (1996). Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Press.
21. DAWSON, D. A. and SALEHI, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141-180.
22. DETTWEILER, E. (1991). Stochastic integration relative to Brownian motion on a general Banach space. Doǧa Mat. 15 6-44.
23. r theorem of the Helmhotz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 685-700.
24. GA̧TAREK, D. and GOŁDYS, B. (1993). On weak solutions of stochastic equations in Hilbert spaces. Stochastics Stochastic Rep. 46 41-51.
25. r spaces. Arch. Ration. Mech. Anal. 89 251-265.
26. GIGA, Y. and SOHR, H. (1989). On the Stokes operator in exterior domain. J. Fac. Sci Univ. Tokyo Sect. IA Math. 36 103-130.
27. IKEDA, N. and WATANABE, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
28. ITÔ, K. (1984). Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. SIAM, Philadelphia.
29. n). Rev. Mat. Iberoamericana 2 73-88.
30. 2). Duke Math. J. 55 487-499.
31. KUO, H. H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Math. 463. Springer, Berlin.
32. KWAPIEŃ, S. and WOYCZYNSKI, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston.
33. LIONS, P. L. (1996). Mathematical Topics in Fluid Mechanics 1. Oxford Science Publishers.
34. METIVIER, M. (1988). Stochastic Partial Differential Equations in Infinite Dimensional Spaces. Scuola Normale Superiore, Pisa.
35. MÉTIVIER, M. and NAKAO, S. (1987). Equivalent conditions for the tightness of a sequence of continuous Hilbert valued martingales. Nagoya J. Math. 106 113-119.
36. NEIDHARDT, A. L. (1978). Stochastic integrals in 2-uniformly smooth Banach spaces. Ph.D. dissertation, Univ. Wisconsin.
37. PARDOUX, E. (1975). Equations aux derivées partielles stochastiques nonlinéaires monotones. Thèse, Univ. Paris XI.
38. PESZAT, S. and ZABCZYK, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Processes Appl. 72 187-204.
39. PRÜSS, J and SOHR, H. (1990). On operators with bounded imaginary powers. Math. Z. 203 429-452.
40. p-spaces. Hiroshima Math. J. 23 161-192.
41. ROZOVSKII, B. L. (1990). Stochastic Evolution Equations. Linear Theory and Applications to Non-linear Filtering. Kluwer, Dordrecht.
42. B. Am. J. Math. 93 299-309.
43. STEIN, E. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press.
44. TAYLOR, M. E. (1981). Pseudodifferential Operators. Princeton Univ. Press.
45. TEMAM, R. (1979). Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam.
46. TRIEBEL, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam.
47. VISHIK, M. I. and FURSIKOV, A. V. (1988). Mathematical Problems of Statistical Hydromechanics. Kluwer, Dordrecht.