Geometric Frustration in Vector Triad Models and a Possible Barrier to Finite-Time Blow-Up in the Navier–Stokes Equations

This paper introduces and formalises the concept of geometric frustration in vector triad models that share the cross-product structure of the three‑dimensional Navier–Stokes nonlinearity. As stated in the abstract, “the nonlinear term cannot inject energy coherently into smaller scales, preventing a self-amplifying cascade,” and this mechanism “provides a unified explanation for the absence of finite-time blow-up in the L’vov vector shell model.” The analysis shows that when the interacting vectors satisfy a mild alignment condition, the forward cascade is suppressed because the triple cross-product structure confines the nonlinear term to the span of the already excited modes. This geometric obstruction contrasts sharply with the scalar GOY model, where perfect phase alignment produces a finite-time singularity. The paper concludes by discussing how similar frustration may operate in the full Navier–Stokes equations.