Martingale Suitable Weak Solutions of $3$-D Stochastic Navier-Stokes Equations with Vorticity Bounds

In this paper, we construct martingale suitable weak solutions for $3$-dimensional incompressible stochastic Navier-Stokes equations with generally non-linear noise. In deterministic setting, as widely known, ``suitable weak solutions'' are Leray-Hopf weak solutions enjoying two different types of local energy inequalities (LEIs). In stochastic setting, we apply the idea of ``martingale solution", avoid transforming to random system, and show new stochastic versions of the two local energy inequalities. In particular, in additive and linear multiplicative noise case, OU-processes and the exponential formulas DO NOT play a role in our formulation of LEIs. This is different to \cite{FR02,Rom10} where the additive noise case is dealt. Also, we successfully apply the concept of ``a.e. super-martingale'' to describe this local energy behavior. To relate the well-known ``dissipative weak solutions" come up with in \cite{DR00}, we derive a local energy equality and extend the concept onto stochastic setting naturally. For further regularity of solutions, we are able to bound the $L^\infty\big([0,T];L^1(\Omega\times\mathbb T^3)\big)$ norm of the vorticity and $L^{\frac{4}{3+\delta}}\big(\Omega\times[0,T]\times\mathbb T^3\big)$ norm of the gradient of the vorticity, in case that the initial vorticity is a finite regular signed measure.