Stochastic 2-D Navier—Stokes Equation

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Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations (2008, background): “However, since we deal with an abstract hydrodynamical model with a forcing term which contains a stochastic control under a minimal set of hypotheses, the argument requires substantial modiﬁcations compared to that of [34] or [27].”
Local martingale and pathwise solutions for an abstract fluids model (2010, methodology): “mam Remark 6.1. As noted in the introduction the Stochastic Navier-Stokes equations have been extensively studied. Initially these equations were considered with an additive noise. See [3] and later, [12, 35]. In this case, a classical transformation, allows one to treat ω ∈ Ω as a parameter in the ”
Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise (2006, methodology): “The stochastic Navier–Stokes equation has been studied by several authors (see for example, Capinsky and Gatarek [5], Flandoli and Gatarek [11], Flandoli and Maslowski [12], and Menaldi and Sritharan [18]) in recent years.”
SPDE in Hilbert space with locally monotone coefficients (2010, methodology): “2) kuk4 L4 ≤ 2kuk 2 L2k∇uk 2 L2, u∈ W 1,2 0 (Λ). Hence combining with (3.1) we have V ∗hA(u)−A(v),u−viV ≤ − 1 2 ku− vk2 V + K+Kkvk2 V ku− vk2 H, u,v∈ V. (2) For d= 3 we use the following estimate (cf.[22]) (3.3) kuk4 L4 ≤ 4kukL2k∇uk 3 L2, u∈ W 1,2 0 (Λ), then the second assertion can be derived simi”