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

TL;DR

This paper proves finite-time blow-up for a class of stochastic fractional Navier-Stokes equations, showing how vortex stretching, temporal memory, and superlinear noise lead to singularity formation.

Contribution

It establishes the first finite-time blow-up results for stochastic fractional Navier-Stokes equations with memory and superlinear noise, revealing conditions for vortex singularity.

Findings

Finite-time blow-up of vorticity in stochastic fractional Navier-Stokes equations.

Vortex stretching combined with noise causes instability and singularity.

Critical memory window allows instability to develop into finite-time blow-up.

Abstract

We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \[ \Caputo \mathbf{u} = -(\mathbf{u} \cdot \nabla)\mathbf{u} - \nabla p + \nu \fLaplacian \mathbf{u} + I^{1-\beta}[\sigma(\mathbf{u}) \dot{W}], \quad \nabla \cdot \mathbf{u} = 0, \] with dissipation $(-\Delta)^{\alpha/2}$ for $\alpha \in (1, 3/2)$, Caputo time-memory $\partial_t^\beta$, and superlinear noise $|\mathbf{u}|^{1+\gamma}$, 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 $\partial_t^\beta$, 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 $|\mathbf{u}|^{1+\gamma}$ noise, guarantees the vorticity will spin up to infinity in finite time.