Global L2-solutions of stochastic Navier–Stokes equations

By R. Mikulevicius, Boris L. Rozovskii. 2005

213 citations recorded by the source corpus.

This paper concerns the Cauchy problem in R d for the stochastic Navier-Stokes equation ∂ 1 u = Δu - (u, ⊇)u - ⊇ p + f(u) + [(σ, ⊇)u - ⊇p + g(u)] o W. u(0) = u 0 , div u = 0, driven by white noise W. Under minimal assumptions on regularity of the coefficients and random forces, the existence of a global weak (martingale) solution of the stochastic Navier-Stokes equation is proved. In the two-dimensional case, the existence and pathwise uniqueness of a global strong solution is shown. A Wiener chaos-based criterion for the existence and uniqueness of a strong global solution of the Navier-Stokes equations is established.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/b3f3e6f1edab833219732d97d1ba80e1cf972cc7

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Citations in the corpus

Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction (2013, background): “Existence and uniqueness have been studied in various articles (se for instance [1], [3], [4], [5], [9], [10], [11], [13], [15], [21], [30], [32], [51], [53], [54], [68], [69]) and this part of the theory is well understood.”
Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains (2012, background): “solutions of the stochastic Navier–Stokes equations driven by white noise in the whole space Rd, (d ≥ 2), are investigated in [24].”
Stochastic Differential Equations: A Wiener Chaos Approach (2005, unknown): “of the Cameron-Martin basis, we can derive the propagator system for coefficientsuα [ 31, Theorem 3.2 ]: Theorem 13.5.”
Strong pathwise solutions of the stochastic Navier-Stokes system (2009, methodology): “[25] R. Mikulevicius and B. L. Rozovskii, Global L2-solutions of stochastic Navier-Stokes equations, Ann.”
An Introduction to 3D Stochastic Fluid Dynamics (2008, unknown): “Relevant references on the topics of this section are, among others, [8], [12], [15], [16], [18], [19], [24], [25], [26], [33], [35], [36], [40], [41], [42], [45], [ 57 ], [58], [59], [61], [63], [64], [66], [67].”
The existence of martingale solutions to the stochastic Boussinesq equations (2011, methodology): “Mikulevicius and Rozovskii [17] proved the existence of a martingale solution of the stochastic Navier-Stokes equations in Rd, d ≥ 2.”
Local and global well-posedness of SPDE with generalized coercivity conditions☆ (2012, background): “Subsequently, Mikulevicius and Rozovskii in [34] showed the existence of martingale solutions to stochastic Navier-Stokes equations in R(d ≥ 2) under weaker assumptions on the coefficients.”
On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier–Stokes equations (2006, background): “Later, this approach was substantially developed and extended by many authors (see, e.g. Mikulevicius & Rozovskii (2004, 2005) and the references therein).”
Local martingale and pathwise solutions for an abstract fluids model (2010, methodology): “with our methods. 1 2 A.Debussche,N.Glatt-HoltzandR.Temam Let us recall at this point that the theory of the related stochastic Navier-Stokesequations haveundergone substantial developments; see e.g. [3, 48, 9, 8, 17, 12, 2, 5, 6, 19, 10, 36, 37, 45, 18, 23]. However to emphasizethe notable diﬀerenc”