Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

By Michael Röckner, Xicheng Zhang. 2008

97 citations recorded by the source corpus.

In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993–1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.

Subfields: Applied Mathematics, Computational Mechanics, Finance

View Paper: https://www.semanticscholar.org/paper/3507a2631c31f1618ba77a6c83a138fe682a26e1

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In Applied Mathematics

1. Xicheng Zhang
2. Tusheng Zhang
3. Michael Röckner
4. Shijie Shang
5. Wei Liu
6. Feng-Yu Wang
7. Longjie Xie
8. Franco Flandoli

In Computational Mechanics

1. Tusheng Zhang
2. Michael Röckner
3. Shijie Shang
4. Xicheng Zhang
5. Paul Dupuis
6. A. Budhiraja
7. Longjie Xie

In Finance

1. Tusheng Zhang
2. Xicheng Zhang
3. Michael Röckner
4. Shijie Shang
5. Wei Liu
6. Feng-Yu Wang
7. A. Budhiraja
8. Paul Dupuis

Citations in the corpus

• Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations (2010, methodology): “It was proved in [21, 20] that under (H1) and (H2), there exists a unique functional Φ : CT (U) → CT (H(1)) (13) such that u (t, ω) := Φ (W (·, ω))(t) satisfies u (·, ω) ∈ L(2)(0, T ; H(2)) for P -almost all ω ∈ Ω (14) and u (t) = u0 + ∫ t”
• Stochastic 3D tamed Navier-Stokes equations: Existence, uniqueness and small time large deviation principles (2012, methodology): “However, since the underlying domain in [26] was all of R3 or the torus, the existence of a strong solution was obtained indirectly via the Yamada–Watanabe Theorem by proving the existence of martingale solutions and pathwise uniqueness.”
• Well-posedness of stochastic partial differential equations with fully local monotone coefficients (2022, methodology): “We refer to [73, 76, 77, 93] and references therein for the early applications of variational approaches to SPDEs.”