Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension d 3: Existence and (non-) triviality

We investigate fractional stochastic Navier-Stokes equations in d 3, driven by the random force (-) ^{2} which, as we show, corresponds to a fractional version of the Landau-Lifshitz random force in the physics literature. We obtain the existence and uniqueness of martingale solutions on the torus Tᵈ for > d2. For 1 the equation is supercritical and we regularize the problem by introducing a Galerkin approximation and we study the large scale behavior of the truncated model on ᵈ. We show that the nonlinear term in the Galerkin approximation vanishes on large scales when < 1 and the model converges to the linearized equation. For = 1 the nonlinear term gives a nontrivial contribution to the large scale beahvior, and we conjecture that the large scale behavior is given by a linear model with strictly larger effective diffusivity compared to simply dropping the nonlinear term. The effective diffusivity is explicitly given in terms of the model parameters.