Orbit-Level Stretching in Cubic Fourier-Galerkin Navier-Stokes: Sharp Incidence, Spectral Decay, and a Continuation Criterion

Abstract

I study orbit-level enstrophy stretching in a cubic Fourier-Galerkin truncation of the three-dimensional incompressible Navier-Stokes equations, reduced by the full octahedral symmetry group $O_h$. The nonlinear transfer compresses to an orbit-level matrix whose symmetric part $V_N$ governs net enstrophy growth. I reduce the stretching problem to an orbit--triad incidence estimate and close it by a face-normalized decomposition and a two-squares argument, establishing the sharp bound \begin{equation} c\,N^3\le\max_\alpha \sum_\beta \sqrt{\Gamma_{\alpha\beta}}\le C\,N^{3}. \end{equation} A weighted-incidence refinement then yields, in the isotropic unit-energy ensemble, \begin{equation} \mathbb{E}\,\rho(V_N)\le C\, N^{-3/2}\to 0, \qquad \mathbb{E}\,\nu_c^*(N)\le C\, N^{-7/2}\to 0, \end{equation} where $\nu_c^*(N)=\rho(V_N)/N^2$ is the orbit-level critical-threshold ratio. For Sobolev-class data with ${ \left\lVert u \right\rVert}_{H^s}\le M$ and $s>2$, a stronger deterministic bound ${{ \left\lVert V_N \right\rVert} }_\infty\le C_s M^3$ holds uniformly in $N$, with $\nu_c^*(N)\to 0$ for all $s>3/2$. A comparison with Tao's averaged Navier-Stokes construction shows that the orbit-level subcriticality is a structural property of the true nonlinearity that is violated by known blowup mechanisms. Monte Carlo experiments at $N=1,\ldots,8$ under both isotropic and Kolmogorov-spectrum ensembles confirm the decay, with $\rho(V_N)\sim N^{-2.6}$ far exceeding the proven upper bound. Tracking these bounds along the Galerkin evolution yields an orbit-level continuation criterion for the strong solution: global regularity holds if and only if $\int_0^T {{ \left\lVert V_N \right\rVert}}_\infty\,dt$ remains bounded uniformly in $N$.