Stochastic Fractional Navier-Stokes Equations: Finite-Time Blow-up for Vortex Stretch Singularities
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arXiv
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| Format: | Preprint |
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2025
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| author | Saucedo, Joel Lamba, Uday |
| author_facet | Saucedo, Joel Lamba, Uday |
| contents | We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \[ \Caputo \mathbf{u} = -(\mathbf{u} \cdot \nabla)\mathbf{u} - \nabla p + ν\fLaplacian \mathbf{u} + I^{1-β}[σ(\mathbf{u}) \dot{W}], \quad \nabla \cdot \mathbf{u} = 0, \] with dissipation $(-Δ)^{α/2}$ for $α\in (1, 3/2)$, Caputo time-memory $\partial_t^β$, and superlinear noise $|\mathbf{u}|^{1+γ}$, proving that for a critical window of memory, $β\in (\fracα{α+3}, β_c(α,γ))$, 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 $\partial_t^β$, 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 $|\mathbf{u}|^{1+γ}$ noise, guarantees the vorticity will spin up to infinity in finite time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_08810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Fractional Navier-Stokes Equations: Finite-Time Blow-up for Vortex Stretch Singularities Saucedo, Joel Lamba, Uday Probability Mathematical Physics Analysis of PDEs We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \[ \Caputo \mathbf{u} = -(\mathbf{u} \cdot \nabla)\mathbf{u} - \nabla p + ν\fLaplacian \mathbf{u} + I^{1-β}[σ(\mathbf{u}) \dot{W}], \quad \nabla \cdot \mathbf{u} = 0, \] with dissipation $(-Δ)^{α/2}$ for $α\in (1, 3/2)$, Caputo time-memory $\partial_t^β$, and superlinear noise $|\mathbf{u}|^{1+γ}$, proving that for a critical window of memory, $β\in (\fracα{α+3}, β_c(α,γ))$, 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 $\partial_t^β$, 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 $|\mathbf{u}|^{1+γ}$ noise, guarantees the vorticity will spin up to infinity in finite time. |
| title | Stochastic Fractional Navier-Stokes Equations: Finite-Time Blow-up for Vortex Stretch Singularities |
| topic | Probability Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2507.08810 |