Martingale suitable weak solutions of 3-D stochastic Navier-Stokes equations with vorticity bounds

Abstract

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 [16], [44] 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 [12], 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 $L^{\infty }\left(\right[0,T];L^1(\Omega \times T^3\left)\right)$-norm of the vorticity and $L^{\frac{4}{3+\delta }}(\Omega \times [0,T]\times T^3)$-norm of the gradient of the vorticity, in case that the initial vorticity is a finite regular signed measure.