Volume 29, Number 4 (2001), 1796-1832.
Stochastic two dimensional euler equations
Zdzislaw Brzezniak and Szymon Peszat
More by Zdzislaw Brzezniak
More by Szymon Peszat
The existence of a martingale solution to 2-dimensional stochastic Euler equations is proved. The constructed solution is a limit as the viscosity converges to zero of a sequence of solutions to modified Navier–Stokes equations.
Ann. Probab., Volume 29, Number 4 (2001), 1796-1832.
First available in Project Euclid: 5 March 2002
MR1880243
Primary: 35Q05: Euler-Poisson-Darboux equations 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 30H15: Nevanlinna class and Smirnov class 60G60: Random fields
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60J25
Stochastic Euler equations stochastic Navier –Stokes equations stochastic hydromechanics martingale problems on Banach spaces
Brzezniak, Zdzislaw; Peszat, Szymon. Stochastic two dimensional euler equations. Ann. Probab. 29 (2001), no. 4, 1796--1832. doi:10.1214/aop/1015345773. https://projecteuclid.org/euclid.aop/1015345773
[1] Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR450957
[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.
Mathematical Reviews (MathSciNet): MR23:A2610
Zentralblatt MATH: 0093.10401
Digital Object Identifier: doi:10.1002/cpa.3160120405
[3] Bardos, C. (1972). Existence et unicit´e de la solution de l'equation d'Euler en dimension deux. J. Math. Anal. Appl. 40 769-790.
Mathematical Reviews (MathSciNet): MR48:11813
Digital Object Identifier: doi:10.1016/0022-247X(72)90019-4
[4] Baxendale, P. (1976). Gaussian measures on function spaces. Amer. J. Math. 98 891-952.
Mathematical Reviews (MathSciNet): MR57:7660
Digital Object Identifier: doi:10.2307/2374035
JSTOR: links.jstor.org
[5] Bergh, J. and L ¨ofstr ¨om, 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.
Mathematical Reviews (MathSciNet): MR2000d:60112
Digital Object Identifier: doi:10.1007/s000300050063
[8] Bessaih, H. (1999). Martingale solutions for stochastic Euler equations. Stochastic Anal. Appl. 17 713-725.
Mathematical Reviews (MathSciNet): MR2001e:60126
Digital Object Identifier: doi:10.1080/07362999908809631
[9] Brze´zniak,(1995). Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4 1-45.
Mathematical Reviews (MathSciNet): MR95m:35213
Digital Object Identifier: doi:10.1007/BF01048965
[10] Brze´zniak,(1997). On stochastic convolutions in Banach spaces and applications. Stochastics Stochastics Rep. 61 245-295.
Mathematical Reviews (MathSciNet): MR1488138
[11] Brze´zniak,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] Brze´zniak,and Ga tarek, D. (1999). Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces. Stochastic Processes Appl. 87 187- 226.
Mathematical Reviews (MathSciNet): MR2000m:60076
Digital Object Identifier: doi:10.1016/S0304-4149(99)00034-4
[13] Brze´zniak,and Peszat, S. (1999). Space-time continuous solutions to SPDEs driven by a homogeneous Wiener process. Studia Math. 137 261-299.
[14] Brze´zniak,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.
Mathematical Reviews (MathSciNet): MR2002c:76035
[15] Capi ´nski, M. and Cutland, N. J. (1999). Stochastic Euler equations on the torus. Ann. Appl. Probab. 9 688-705.
Mathematical Reviews (MathSciNet): MR2000i:35231
Digital Object Identifier: doi:10.1214/aoap/1029962809
Project Euclid: euclid.aoap/1029962809
[16] Capi ´nski, M. and Ga tarek, D. (1994). Stochastic equations in Hilbert space with applications to 3-D stochastic Navier-Stokes equations. J. Funct. Anal. 126 25-35.
[17] Capi ´nski, 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.
Mathematical Reviews (MathSciNet): MR95g:60073
[20] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR97k:60165
[21] Dawson, D. A. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141-180.
Mathematical Reviews (MathSciNet): MR82c:60102
[22] Dettweiler, E. (1991). Stochastic integration relative to Brownian motion on a general Banach space. Do ga Mat. 15 6-44.
Mathematical Reviews (MathSciNet): MR93b:60112
[23] Fujiwara, D. and Morimoto, H. (1977). An Lr theorem of the Helmhotz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 685-700.
[24] Ga tarek, D. and Goldys, B. (1993). On weak solutions of stochastic equations in Hilbert spaces. Stochastics Stochastic Rep. 46 41-51.
[25] Giga, Y. (1985). Domains of fractional powers of the Stokes operator in Lr 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] It o, K. (1984). Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. SIAM, Philadelphia.
[29] Kato, T. and Ponce, G. (1986). Well-posedness of the Euler and Navier-Stokes equations in Lebesgue spaces Lps Rn. Rev. Mat. Iberoamericana 2 73-88.
[30] Kato, T. and Ponce, G. (1987). On nonstationary flows of viscous and ideal fluid in Lps R2. Duke Math. J. 55 487-499.
Digital Object Identifier: doi:10.1215/S0012-7094-87-05526-8
Project Euclid: euclid.dmj/1077306162
[31] Kuo, H. H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Math. 463. Springer, Berlin.
[32] Kwapie ´n, S. and Woyczynski, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkh¨auser, 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.
Mathematical Reviews (MathSciNet): MR90a:60112
[35] M´etivier, M. and Nakao, S. (1987). Equivalent conditions for the tightness of a sequence of continuous Hilbert valued martingales. Nagoya J. Math. 106 113-119.
Project Euclid: euclid.nmj/1118780705
[36] Neidhardt, A. L. (1978). Stochastic integrals in 2-uniformly smooth Banach spaces. Ph.D. dissertation, Univ. Wisconsin.
[37] Pardoux, E. (1975). Equations aux deriv´ees partielles stochastiques nonlin´eaires monotones. Th ese, 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] Pr ¨uss, J and Sohr, H. (1990). On operators with bounded imaginary powers. Math.203 429-452.
[40] Pr ¨uss, J. and Sohr, H. (1993). Imaginary powers of elliptic second order differential operators in Lp-spaces. Hiroshima Math. J. 23 161-192.
Mathematical Reviews (MathSciNet): MR94d:47051
Project Euclid: euclid.hmj/1206128381
[41] Rozovskii, B. L. (1990). Stochastic Evolution Equations. Linear Theory and Applications to Non-linear Filtering. Kluwer, Dordrecht.
[42] Seeley, R. (1971). Norms and domains of complex powers AzB. 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. NorthHolland, Amsterdam.
[47] Vishik, M. I. and Fursikov, A. V. (1988). Mathematical Problems of Statistical Hydromechanics. Kluwer, Dordrecht.
Stochastic Euler equations on the torus
Capiński, Marek and Cutland, Nigel J., The Annals of Applied Probability, 1999
Viscous Limits to Piecewise Smooth Solutions for the Navier-Stokes Equations of One-dimensional Compressible Viscous Heat-conducting Fluids
Ma, Shixiang, Methods and Applications of Analysis, 2009
Vanishing viscosity and the accumulation of vorticity on the boundary
Kelliher, J. P., Communications in Mathematical Sciences, 2008
From compressible to incompressible inhomogeneous flows in the case of large data
Danchin, Raphaël and Mucha, Piotr Bogusław, Tunisian Journal of Mathematics, 2019
A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations
Gallagher, Isabelle, Journal of Mathematics of Kyoto University, 2000
Global L2-solutions of stochastic Navier–Stokes equations
Mikulevicius, R. and Rozovskii, B. L., The Annals of Probability, 2005
On the global well-posedness of 3-d Navier-Stokes equations with vanishing horizontal viscosity
Abidi, Hammadi and Paicu, Marius, Differential and Integral Equations, 2018
On the convergence of Euler-Stokes splitting of the Navier-Stokes equations
Popa, Cătălin, Differential and Integral Equations, 2002
Stability and instability of Navier boundary layers
Paddick, Matthew, Differential and Integral Equations, 2014
Équation anisotrope de Navier-Stokes dans des espaces critiques
Paicu, Marius, Revista Matemática Iberoamericana, 2005