We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \ u = - (u) u - p + ν u + I^1-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.
Saucedo et al. (Wed,) studied this question.