The Navier–Stokes Equations: A Critical Review of Theory, Computation, Turbulence, and Data-Driven Frontiers

Abstract

The Navier–Stokes equations describe the motion of viscous fluids and stand at the center of modern fluid mechanics. They also define one of the hardest open problems in mathematics, for the question of whether smooth three-dimensional solutions can develop singularities in finite time remains unresolved. This review provides a critical, evidence-based synthesis of the subject from its foundations in the work of Leray and Hopf through the mathematical, computational, and data-driven landscape of 2026. We first examine the equations and their physical content, then trace the theory of well-posedness: weak and strong solutions, regularity criteria, partial regularity, uniqueness, and nonuniqueness. We appraise what is known about possible singularities and the energy and attractor theory that governs long-time behavior. The statistical description of turbulence, from Kolmogorov scaling to anomalous dissipation, is then reviewed alongside its unresolved tensions. We evaluate the numerical toolbox — finite difference, projection, finite element, spectral, discontinuous Galerkin, lattice Boltzmann, and finite volume methods — with attention to stability, accuracy, and the pressure-velocity coupling that complicates incompressible computation. Modeling hierarchies from direct numerical simulation through large-eddy and Reynolds-averaged closures are assessed, as are physics-informed neural networks and machine-learning-accelerated solvers that have reshaped the field since 2019. Applications in cardiovascular hemodynamics, geophysical flows, multiphase systems, combustion, and viscoelastic fluids illustrate both achievement and persistent gap. We close with a concrete research agenda, emphasizing the convergence of mathematical proof, high-resolution computation, and physical measurement. Keywords: Navier–Stokes equations; weak solutions; regularity; turbulence; direct numerical simulation; large-eddy simulation; physics-informed neural networks; hemodynamics; Reynolds number.