Convex Integration and Non-Uniqueness of Weak Solutions to the 3D Navier-Stokes Equations

This monograph explores the recent breakthrough in the analysis of the three-dimensional incompressible Navier-Stokes equations, specifically concerning the non-uniqueness of weak solutions. While the existence of global weak solutions (Leray-Hopf) has been established since 1934, their uniqueness remains one of the most challenging open problems in mathematical fluid dynamics. We detail the application of the convex integration method, originally developed by Nash and Gromov for isometric embeddings and adapted to the Euler equations by De Lellis and Székelyhidi, to the viscous setting. We present the construction of "wild" weak solutions that possess finite kinetic energy but violate the energy equality, exhibiting anomalous dissipation. The iterative scheme relies on the introduction of intermittent Beltrami flows and a careful handling of the Reynolds stress tensor to overcome the viscous smoothing. These results demonstrate that the class of distributional weak solutions is too broad to ensure uniqueness, thereby sharpening the boundaries of the regularity theory for the Navier-Stokes system.