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In this paper, we construct suitable weak martingale solutions for 3-dimensional incompressible stochastic Navier-Stokes equations with linear multiplicative noise. In deterministic setting, as widely known, ``suitable weak solutions'' are Leray-Hopf weak solutions enjoying two different types of local energy inequalities. In stochastic setting, we are able to show corresponding stochastic versions of the two local energy inequalities, where a semi-martingale term occurs as the effect of the noise. Note that the exponential formulas, which are widely used to transform the stochastic PDEs system into a random one, DO NOT show up in our formulations of local energy inequality. This is different to FR02, Rom10 where the additive noise case is dealt. Also, a path-wise result in terms of ``a. e. super-martingale'' is derived from the stochastic local energy inequalities. Further more, if the initial vorticity is a finite Radon measure, we are able to bound the L¹ (;L^ (0, T;L¹) ) norm of the vorticity and L^4{3+} (0, T T³) norm of the gradient of the vorticity.
Chen et al. (Sun,) studied this question.