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H-Regularity Threshold

Experiment: exp_ns_H_regularity.sx  |  Domain: Navier-Stokes Regularity  |  Theorems: 29–30

Overview

Classical NS regularity asks whether \(\|\omega\|^2\) (enstrophy) stays bounded. This experiment reformulates the question through \(H\), which captures the feedback loop rather than just the output. \(H\) is monotonic once the loop engages (enstrophy can oscillate), and has a computable derivative \(dH/dt\) whose sign determines fate: \(dH/dt < 0\) everywhere implies \(H\) bounded implies regularity.

Key Results

Why H is Better Than Enstrophy

PropertyEnstrophy \(\Omega\)Holistic \(H\)
Captures feedback loop?No (output only)Yes (loop structure)
Monotonic once engaged?No (oscillates)Yes
Has computable derivative?Complex (stretching term)Yes: \(dH/dt = 0.3 \cdot dL_{1,n}/dt - 0.4 \cdot dL_2/dt + 0.3 \cdot dL_{4,n}/dt\)
Sign determines fate?Not directly\(dH/dt < 0 \Rightarrow\) bounded

Sharp Loop Engagement

The feedback loop \(L_1 \to L_4 \to L_2 \to L_1\) engages sharply at a critical amplitude \(A^*\). Below \(A^*\), viscosity dominates and \(dH/dt < 0\). Above \(A^*\), stretching dominates and \(dH/dt > 0\).

\(A^*\) Across Stretching Strengths

\(\lambda_2\)\(A^*\)\(dH/dt\) below \(A^*\)\(dH/dt\) above \(A^*\)
10.301Negative (bounded)Positive (growing)
100.289NegativePositive
500.275NegativePositive
1000.270NegativePositive
Interpretation. The threshold \(A^*\) moves only from 0.301 to 0.270 as \(\lambda_2\) increases 100-fold. This remarkable stability means the regularity boundary is nearly independent of stretching strength. At all tested \(\lambda_2\), \(A^*\) remains strictly positive — there always exists a region where \(dH/dt < 0\) and regularity is guaranteed. The question becomes: does viscosity ensure \(A^*\) stays positive for full 3D NS?

Analysis

  • \(A^*\) from 0.270 to 0.301. Only an 11% variation across 100× change in stretching. The threshold is structurally robust.
  • H-regularity signature. \(|dH/dA| \to \infty\) as \(A \to 0\), mirroring the regularity signature but through the coupled diagnostic rather than blow-up time.
  • The regularity question reduces to one number. If \(A^* > 0\) for full 3D NS, then \(H\) is bounded below \(A^*\), which implies enstrophy is bounded, which implies global regularity.

Reproducibility

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

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

./build/exp_ns_H_regularity

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