Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

By Igor D. Chueshov, Annie Millet. 2008

256 citations recorded by the source corpus.

We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also some shell models of turbulence. We state the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by a weak convergence method.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/83bf7245f4b5853ae1a0186c26efa5004679af7c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Wei Liu
2. Michael Röckner
3. Tusheng Zhang
4. Shijie Shang
5. Wei David Liu
6. Zdzisław Brzeźniak
7. Jiahui Zhu
8. Paul Dupuis

In Control and Systems Engineering

1. Wei Liu
2. Tusheng Zhang
3. Shijie Shang
4. Michael Röckner
5. Wei David Liu
6. Jiahui Zhu
7. Zdzisław Brzeźniak
8. Jiang Chen

In Finance

1. Tusheng Zhang
2. Wei Liu
3. Michael Röckner
4. A. Budhiraja
5. Paul Dupuis
6. Shijie Shang
7. Jiang Chen
8. Wei David Liu

Citations in the corpus

• Local and global well-posedness of SPDE with generalized coercivity conditions☆ (2012, background): “Although we have quite satisfactory results for 2D stochastic Navier-Stokes equations such as well-posedness, small noise asymptotics and ergodicity (cf.[31, 13, 24] and the references therein), the results for the three dimensional case are still quite incomplete due to the lack of uniqueness (cf.”
• Large Deviations for Stochastic Partial Differential Equations Driven by a Poisson Random Measure (2012, methodology): “In recent years an approach based on certain variational representation formulas for moments of nonnegative functionals of Brownian motions [7] has been increasingly used for the study of the small noise large deviation problem for Brownian motion driven infinite dimensional systems [2, 7, 8, 10, 13”
• Well-posedness of stochastic partial differential equations with Lyapunov condition (2013, background): “Recently, this framework has been substantially extended by Röckner and the author in [23] for more general class of SPDE with coefficients satisfying the coercivity and local monotonicity conditions, hence many more fundamental examples such as stochastic Burgers type equations and stochastic 2D Na”
• Variational representations for continuous time processes (2011, methodology): “For example, Brownian motion representations have been used by [1, 6, 7, 8, 10, 11, 18, 19, 21, 22, 24, 25, 26, 29, 30] in the large deviation analysis of solutions to SPDEs in the small noise limit, and recently in [5] for interacting particle limits.”
• Strong solutions for SPDE with locally monotone coefficients driven by L\'{e}vy noise (2011, methodology): “2) in [13] implies the coercivity (H3) holds, and the local monotonicity (H2) follows from (2.”
• Splitting up method for the 2D stochastic Navier–Stokes equations (2013, unknown): “4 in [9] can be supposed to be weak-star convergent to uh in L (Ω, L(0, T ;H)).”
• Well-posedness of stochastic partial differential equations with fully local monotone coefficients (2022, methodology): “see [48, 27, 5] for regularity of solutions, [18, 74, 47, 19, 88, 52, 61, 87, 37, 38] for large deviation principle, [31, 32, 33, 78, 65, 4, 90] for numerical approximation, [84, 46, 85, 86] for the Harnack inequality, gradient estimates, invariant measure and ergodicity, [28, 29] for random attract”