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 $$\text{Caputo}\mathbf{u}=-(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}-\mathrm{\nabla }p+\nu \text{fLaplacian}\mathbf{u}+I^{1-\beta }[\sigma (\mathbf{u})\dot{W}],\mathrm{\nabla }\cdot \mathbf{u}=0,$$ with dissipation $(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$ for $\alpha \in (1,3\mathrm{/}2)$, Caputo time-memory ${\mathrm{\partial }}_t^{\beta }$, and superlinear noise $\mathrm{\mid }\mathbf{u}{\mathrm{\mid }}^{1+\gamma }$, proving that for a critical window of memory, $\beta \in (\frac{\alpha }{\alpha +3},{\beta }_c(\alpha ,\gamma ))$, 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 ${\mathrm{\partial }}_t^{\beta }$, 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 $\mathrm{\mid }\mathbf{u}{\mathrm{\mid }}^{1+\gamma }$ noise, guarantees the vorticity will spin up to infinity in finite time.