## Electronic Communications in Probability

### A Reflection Type Problem for the Stochastic 2-D Navier-Stokes Equations with Periodic Conditions

#### Abstract

We prove the existence of a solution for the Kolmogorov equation associated with a reflection problem for 2-D stochastic Navier-Stokes equations with periodic spatial conditions and the corresponding stream flow in a closed ball of a Sobolev space of the torus $T^2$.

#### Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 29, 304-313.

Dates
Accepted: 21 June 2011
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465261985

Digital Object Identifier
doi:10.1214/ECP.v16-1633

Mathematical Reviews number (MathSciNet)
MR2819654

Zentralblatt MATH identifier
05946901

Rights

#### Citation

Barbu, Viorel; Da Prato, Giuseppe; Tubaro, Luciano. A Reflection Type Problem for the Stochastic 2-D Navier-Stokes Equations with Periodic Conditions. Electron. Commun. Probab. 16 (2011), paper no. 29, 304--313. doi:10.1214/ECP.v16-1633. https://projecteuclid.org/euclid.ecp/1465261985

#### References

• S. Albeverio and A.B. Cruzeiro. Global flows with invariant (Gibbs) measure for Euler and Navier-Stokes two dimensional fluids. Comm. Math. Phys. 129 (1990), 431-444.
• S. Albeverio, M. Ribeiro De Faria and R. Høegh-Krohn. Stationary measures for the periodic Euler flows in two dimensions. J. Stat. Phys. 20 (1979), 584-595.
• S. Albeverio and B. Ferrario. Uniqueness results for the generators of the two dimensional Euler and Navier-Stokes flows. J. Funct. Anal. 193 (2002), 73-93.
• S. Albeverio, V. Barbu and B. Ferrario. Uniqueness of the generators of the 2-D Euler and Navier-Stokes flows. Stoch. Processes and their Appl. 118 (2008), 2071-2084.
• V. Barbu, G. Da Prato and L. Tubaro. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space. Ann. Probab, 37, n.4, 1427-1458, 2009.
• V. Barbu, G. Da Prato and L. Tubaro. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II. Ann. Inst. Poincaré (to appear).
• F. Cipriano. The two dimensional Euler equations: a statistical study. Comm. Math. Phys. 201 (1999), 139-154.
• G. Da Prato and A. Debussche. Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196 (2002), 180-210.
• F. Flandoli and F. Gozzi. Kolmogorov equation associated to a stochastic Navier-Stokes equation. J. Funct. Anal. 160 (1998), 312-336.
• D. Nualart and E. Pardoux. White noise driven quasilinear SPDE's with reflection. Prob. Th. and Related Fields 93, 77-89, 1992.
• W. Stannat. A new apriori estimates for the Kolmogorov operator of a 2-D stochastic Navier-Stokes equation. Infinite Dimensional Anal. Quantum Probab. Related Topics, 10 4 (2007), 483-497.