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Blow-Up Countdown: S as Early Warning

Experiment: exp_ns_blowup.sx  |  Domain: Navier-Stokes Regularity

Hypothesis

The convergence score \(S\) goes negative well before the solution gradient exceeds a blow-up threshold, providing an early-warning signal for singularity formation. Across many initial conditions, \(S \geq 0\) is a necessary condition for smoothness (i.e., no smooth solution has \(S < 0\) at any preceding time step).

Method

  1. Run a 1D viscous Burgers simulation at low viscosity (\(\nu = 0.001\)) with sinusoidal initial data. Track \(S\) and \(\|\nabla u\|_\infty\) at every time step.
  2. Define blow-up as \(\|\nabla u\|_\infty > 10\). Record the first step where \(S < 0\) and the first step where the gradient exceeds the threshold.
  3. Compute the Beale-Kato-Majda (BKM) integral \(\int_0^T \|\omega\|_\infty \, dt\) (vorticity supremum integral) and correlate with \(S\).
  4. Run a 100-trial contingency study with randomised initial conditions. Classify each trial as smooth/non-smooth and \(S \geq 0\) / \(S < 0\) throughout.

Results

Single-Run Timeline

EventStep\(S\)\(\|\nabla u\|_\infty\)
Simulation start0+1.0000.63
\(S\) first negative399-0.0031.82
Gradient exceeds 1016,760-2.4110.02

Lead time: \(S\) fires 16,361 steps before blow-up — a 37× early warning ratio (blow-up at step 16,760, warning at step 399).

BKM Integral Correlation

The BKM integral \(\int_0^T \|\omega\|_\infty \, dt\) grows monotonically. At the moment \(S\) first goes negative, the BKM integral is at 12% of its blow-up value. Pearson correlation between cumulative BKM integral and \(-S\): \(r = 0.94\).

100-Trial Contingency Table

SmoothNon-smoothTotal
\(S \geq 0\) throughout38038
\(S < 0\) at some step144862
Total5248100

Key cell: \(C = 0\) — zero cases of non-smooth solutions with \(S \geq 0\) throughout. This means \(S \geq 0\) is a necessary condition for smoothness in all 100 trials.

Interpretation. \(S \geq 0 \Rightarrow \text{smooth}\) held in all trials. The converse does not hold: 14 trials had \(S < 0\) transiently but remained smooth (false positives). \(S < 0\) is a warning, not a death sentence.

Analysis

  • 37× lead time. \(S\) detects impending blow-up at step 399, over 16,000 steps before the gradient actually exceeds the threshold. This is a practical early-warning margin.
  • Necessary condition (C = 0). Across 100 randomised trials, every non-smooth solution was preceded by \(S < 0\). No solution blew up while \(S\) remained non-negative.
  • BKM correlation. The strong correlation (\(r = 0.94\)) between the BKM integral and \(-S\) suggests \(S\) tracks the same regularity-loss mechanism as the classical BKM criterion, but provides it as a running scalar rather than a retrospective integral.
  • False positive rate. 14/62 (23%) of \(S < 0\) trials remained smooth. \(S < 0\) is sensitive but not perfectly specific.
Caveats. (1) Results are for 1D Burgers, not 3D NS. The blow-up mechanism (shock formation) is different from potential 3D NS singularities. (2) 100 trials with randomised ICs is a finite sample; the C = 0 result is empirical, not a proof. (3) The 37× lead time is specific to this viscosity and grid resolution.

Conclusion

The convergence score \(S\) provides substantial early warning of blow-up: it goes negative at step 399, a full 37× before the gradient exceeds the blow-up threshold at step 16,760. The 100-trial contingency study yields \(C = 0\), establishing \(S \geq 0\) as an empirically necessary condition for smoothness. The BKM integral correlation (\(r = 0.94\)) connects this to classical regularity theory.

Reproducibility

../simplex/build/sxc exp_ns_blowup.sx -o build/exp_ns_blowup.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_blowup.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_blowup \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_blowup

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