3D NS BKM Infrastructure: First Certifying Pseudospectral Blowup-Monitor Runs

The Finding

We extended the 2D BKM diagnostic to three-dimensional vorticity-form incompressible Navier–Stokes with explicit vortex stretching $(\boldsymbol{\omega}\cdot\nabla)\mathbf{u}$ — the mechanism absent in 2D and central to Beale–Kato–Majda blowup criteria in 3D.

Each run tracks max vorticity $\|\omega\|_{L^\infty}$, enstrophy, and the cumulative BKM integral

$\int_0^T \|\omega(\cdot,t)\|_{L^\infty}\, dt$

on a single NVIDIA RTX 5090 using a custom CUDA + cuFFT pseudospectral kernel.

Run	Grid	$\nu$	$\Delta t$	IC	Steps	Result
Smoke	64³	0.01	0.002	Taylor–Green	200	BKM ≈ 1.63; max $\|\omega\| \approx 4.17$ at $t=0.4$
Standard	128³	1e-3	0.002	Random blob	1000	BKM ≈ 1.24; max $\|\omega\| \approx 0.614$ at $t=2.0$
Blowup search	256³	1e-4	0.001	Random blob	500	BKM ≈ 0.44; max $\|\omega\| \approx 0.878$ at $t=0.5$; 2.3 steps/s
Blowup search (long)	256³	1e-4	0.001	Random blob	2000	BKM ≈ 1.76; max $\|\omega\| \approx 0.887$ at $t=2.0$; 880 s
Blowup search (5000)	256³	1e-4	0.001	Random blob	5000	BKM ≈ 4.45; max $\|\omega\| \approx 0.903$ at $t=5.0$; 2190 s
Taylor–Green	256³	1e-3	0.001	Taylor–Green	1000	BKM ≈ 4.23; max $\|\omega\| \approx 4.44$ at $t=1.0$
Kerr (ν sweep)	256³	1e-4	0.001	Kerr tubes	2000	BKM ≈ 9.99; max $\|\omega\| \approx 5.0$; 5.7× random BKM
Kerr (ν sweep)	256³	1e-3	0.001	Kerr tubes	1000	BKM ≈ 4.99 vs random 0.88 at $t=1.0$

Both runs: zero NaN/Inf (exit certificate). No finite-time blowup signal at these resolutions and Reynolds numbers.

Method

Governing equation on a periodic cube:

$\partial_t \omega + (\mathbf{u}\cdot\nabla)\omega = (\boldsymbol{\omega}\cdot\nabla)\mathbf{u} + \nu \nabla^2 \omega$

Velocity from Fourier space: $\hat{\mathbf{u}} = i(\mathbf{k}\times\hat{\boldsymbol{\omega}})/\|\mathbf{k}\|^2$. 2/3 Orszag dealiasing, RK4, fp64 throughout.

Hardware. Physical RTX 5090 (32 GB, Blackwell, CC 12.0), compiled with -arch=sm_120.

Limitations. No spatial convergence study; moderate Re only; 512³ OOM on 32 GB RTX 5090 (grid must be power of 2); 256³ is far below resolutions used in state-of-the-art 3D blowup-search DNS (e.g. Kerr 1993).

References

• Beale, Kato, Majda (1984) — BKM blowup criterion
• Orszag (1971) — 2/3 dealiasing rule
• Canuto et al. (1988); Rogallo (1981) — pseudospectral methods
• Kerr (1993) — landmark 3D Euler blowup-search DNS

Claim validation

Claim	Status	Evidence
3D vortex-stretching DNS runs	Valid infrastructure	CSV logs, certifying exit code
BKM integral monitoring	Implemented	Cumulative $\int_0^t \|\omega\|_\infty ds$ each step
Finite-time blowup	Not claimed	Vorticity bounded at tested $(N,\nu,T)$
Resolution of blowup candidates	Not claimed	128³ certifying sweep only

What we do not claim

• Evidence of Navier–Stokes singularity or blowup
• Sufficient resolution to resolve inertial-range turbulence or blowup-scale structures
• Comparison to published 3D benchmark DNS at identical Re

Phase 5: Kerr-type IC and viscosity sweep

We added a Kerr-class antiparallel vortex-tube initializer (z-aligned Gaussian filaments at $x = \pi \pm 0.55$, Crow-type 3D perturbation) and ran a 256³ viscosity sweep:

IC	$\nu$	$T$	BKM	max $\|\omega\|_\infty$
Kerr	$10^{-3}$	1.0	4.99	5.0
Random	$10^{-3}$	1.0	0.88	0.88
Kerr	$10^{-4}$	2.0	9.99	5.0
Random	$10^{-4}$	2.0	1.76	0.89
Kerr	$10^{-5}$	1.0	4.99	5.0

Structured Kerr IC drives ~5–6× higher BKM than random blob at matched $(N, \nu, T)$ — better aligned with blowup-search literature (Kerr 1993). Still zero NaN/Inf; vorticity bounded at tested resolution. HF configs kerr_nu1e-03, kerr_nu1e-04, kerr_nu1e-05.

Phase 4: targeted blowup search

We ran 256³ random-IC sweeps at $\nu = 10^{-4}$, $\Delta t = 0.001$: 500-, 2000-, and 5000-step certifying runs (BKM ≈ 4.45 by $t=5.0$). A 256³ Taylor–Green benchmark at $\nu = 10^{-3}$ reached BKM ≈ 4.23 by $t=1.0$. Vorticity remains bounded in all runs — zero NaN/Inf. 512³ exceeds 32 GB VRAM on RTX 5090. Data in Hugging Face configs blowup_search, blowup_search_long, blowup_search_5000, and taylor_green_256.

Data: Hugging Face cahlen/cfd-ns3d-bkm · Experiment · Phase 2 finding

Reproduction

git clone https://github.com/cahlen/idontknow.git
cd idontknow
./scripts/experiments/cfd-ns3d-bkm/run.sh 64 0.01 200 0.002 taylor-green
./scripts/experiments/cfd-ns3d-bkm/run.sh 128 0.001 1000 0.002 random
./scripts/experiments/cfd-ns3d-bkm/run.sh 256 0.0001 500 0.001 random
./scripts/experiments/cfd-ns3d-bkm/run.sh 256 0.0001 2000 0.001 random
./scripts/experiments/cfd-ns3d-bkm/run.sh 256 0.0001 5000 0.001 random
./scripts/experiments/cfd-ns3d-bkm/run.sh 256 0.001 1000 0.001 taylor-green
./scripts/experiments/cfd-ns3d-bkm/run.sh 256 0.001 1000 0.001 kerr
./scripts/experiments/cfd-ns3d-bkm/run.sh 256 0.0001 2000 0.001 kerr
./scripts/experiments/cfd-ns3d-bkm/run_phase5_kerr_sweep.sh   # full ν sweep

Human–AI collaboration. Silver-certified (2026-05-31). All code open for verification.