On the Regularity of the Two-Dimensional Navier-Stokes Equations

The global regularity of solutions to the two-dimensional (2D) incompressible Navier-Stokes equations (NSE) is disproved by demonstrating the existence of singularities in plane Couette flow. Using tools from the Sobolev space theory, the energy dissipation analysis, and the energy gradient theory (Dou, 2025), it is shown that the flow develops "velocity discontinuities" under generic disturbances, where the velocity is not differentiable, forming singularities of the Navier-Stokes equation. Thus, the solution fails to be in C^1(OmegaX[0, T )) for any T > T0 (with T0 is the singularity formation time). Specifically, it is demonstrated that such singularities force the solution to exit the Sobolev space H^1(Omega), violating the smoothness criterion required for global regularity. The result of present study indicates that the classical result of Ladyzhenskaya (1969) on the existence of global smooth solutions to the 2D NSE is incorrect.