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In this paper we introduce a probabilistic approach to show the existence ofinitial data with arbitrarily large $L^2(\mathbb{R}^3)$,$\dot{H}^{1/2}(\mathbb{R}^3)$ and $\mathcal{PM}^2$-norms for which aGeneralized Navier-Stokes system generate a global regular solution. Moreprecisely, we show that from a certain family of possible large initial datamost of them give raise to global regular solutions to a given GeneralizedNavier-Stokes system.

The Probabilistic Method and large initial data for Generalized Navier-Stokes systems