On the fractional stochastic Navier-Stokes equations on the torus and on bounded domains

In this work, we introduce and study the well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on bounded domains and on the torus (Briefly dD-FSNSE). We prove the existence of a martingale solution for the general regime. We establish the uniqueness in the case a martingale solution enjoys a condition of Serrin's type on the fractional Sobolev spaces. If an $L^2-$ local weak (strong in probability) solution exists and enjoys conditions of Beale-Kato-Majda type, this solution is global and unique. These conditions are automatically satisfied for the 2D-FSNSE on the torus if the initial data has $H^1-$regularity and the diffusion term satisfies growth and Lipschitz conditions corresponding to $H^1-$spaces. The case of 2D-FSNSE on the torus is studied separately. In particular, we established thresholds for the global existence, uniqueness, space and time regularities of the weak (strong in probability) solutions in the subcritical regime.

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