Stochastic Fractional Navier-Stokes Equations: Finite-Time Blow-up for Vortex Stretch Singularities

We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \[ u = -(u · ∇)u - ∇ p + ν u + I1-β[σ(u) Ẇ], ∇ · u = 0, \] with dissipation (-Δ)α/2 for α∈ (1, 3/2), Caputo time-memory ∂ₜ^β, and superlinear noise |u|1+γ, proving that for a critical window of memory, β∈ (αα+3, βc(α,γ)), the second moment of the vorticity supremum explodes due to a vortex-stretching-driven renewal inequality. This work reveals that when a fluid's temporal memory, governed by ∂ₜ^β, is short enough to permit instability but long enough for that instability to mature, the relentless self-amplification from vortex stretching, when coupled with explosive stochastic kicks from the |u|1+γ noise, guarantees the vorticity will spin up to infinity in finite time.