Navier–Stokes Regularity via Critical Norm Tracking

Formulate the 3D incompressible Navier–Stokes equations on a 2π-periodic domain with divergence-free initial data. Derive the standard energy inequality and identify the Prodi–Serrin regularity condition (2/p + 3/q = 1, q > 3). Implement a pseudo-spectral solver (dealised 2/3 rule) and run parameter sweeps for Reynolds numbers Re ∈ {200, 400, 800, 1600}. Compute time series of \|u\|_{L^2}, \|∇u\|_{L^2}, and the critical norms L^p_t L^q_x for representative (p,q) pairs. Validate the solver against a reference dataset such as JHTDB isotropic turbulence if accessible. Report energy spectra E(k), Kolmogorov scaling windows, and verify numerical stability/convergence. The goal is a proof of the Millennium problem, and a rigorous computational study of regularity indicators and their limitations.