Stability and Well-Posedness of Fractional Navier–Stokes with Directional Fractional Diffusion

We investigate the three-dimensional incompressible fractional Navier–Stokes system with directional fractional diffusion: a vertical dissipative operator of order 2α∈(0,2] acting on the full velocity field together with a horizontal fractional operator of order 2β∈(0,2] applied to the vertical average of one horizontal component. This anisotropic, nonlocal structure captures media in which smoothing acts with unequal strength by direction. For small, divergence-free initial data in an anisotropic energy class, we establish global well-posedness and stability of the zero state, including uniqueness and continuous dependence on the data. The analysis crucially relies on an average oscillation decomposition in the vertical variable, a fractional Poincaré inequality aligned with the vertical direction, and sharp product/commutator bounds compatible with the anisotropic splitting. We provide explicit estimates for direction-dependent smoothing and algebraic decay governed by (α,β), and we show that the conclusions persist under small perturbation of the dissipation parameters.