"Regularity Criteria for the 3D Navier–Stokes Equations: An Analysis of the Beale–Kato–Majda and Prodi–Serrin Conditions

This work presents a detailed mathematical study of two cornerstone regularity criteria for the three-dimensional incompressible Navier–Stokes equations — the Beale–Kato–Majda (BKM) criterion and the Prodi–Serrin conditions. Both provide rigorous bounds under which smooth solutions persist, offering valuable insight into one of the most challenging open problems in mathematical physics: the global regularity versus finite-time blow-up question.The article derives each criterion from the Navier–Stokes system, highlights their scaling properties, and examines the interplay between vorticity growth, velocity integrability, and vortex stretching mechanisms. It also discusses computational implications, particularly for turbulence modeling and singularity detection in numerical simulations.This analysis contributes to the broader understanding of singularity formation mechanisms in fluid dynamics and aligns with ongoing efforts to resolve the Navier–Stokes Millennium Prize Problem. The results are relevant to researchers in mathematical physics, applied mathematics, computational fluid dynamics, and related areas.<br>