Navier-Stokes Regularity and Turbulence

The question of whether smooth solutions to the $3$D incompressible Navier-Stokes equations $\partial_t u + (u \cdot \nabla)u = nuDelta u - \nabla p$, $\nabla \cdot u = 0$ can develop singularities \in finite time remains one of the central open problems \in mathematical physics. We review the classical energy inequality $\frac{1}{2}\frac{d}{dt}|u|_{L^2}^2 + \nu|\nabla u|_{L^2}^2 \leq 0$ and the Caffarelli-Kohn-Nirenberg \partial regularity theorem, which bounds the Hausdorff dimension of the singular set. We then discuss connections to Kolmogorov's statistical theory of turbulence and the $-5/3$ energy spectrum law, and outline recent approaches using convex integration techniques.