Non-Existence of Global Smooth Solutions to the Three-Dimensional Navier-Stokes Equations for Plane Couette Flow

This study investigates the regularity of the three-dimensional (3D) incompressible Navier-Stokes equations (NSE) for plane Couette flow, a canonical shear-driven flow model with a well-defined laminar-turbulent transition threshold. Employing Sobolev space theory and the Poincaré inequality, we rigorously prove that no global smooth solutions exist as the Reynolds number exceeds the critical value \( Re_{cr} \). Prior studies have revealed that a zero velocity gradient on the velocity profile is the necessary and sufficient condition for turbulence generation in 3D plane Couette flow, yet they lack mathematical theoretical proof from the perspective of partial differential equation framework. This study fills this gap via velocity decomposition and singularity analysis. We show that nonlinear disturbance amplification induces local cancellation of mean and disturbance velocity gradients, triggering finite-time singularity formation in flow field, which leads to the breakdown of regularity of the 3D NSE and thus the non-existence of global smooth solutions. It is emphasized that the non-existence of smooth solutions is due to the local regularity breakdown of solutions instead of the velocity blow-up. Further, it is important that the critical condition for regularity breakdown obtained through Sobolev space analysis accords with the critical condition for turbulence onset obtained through experiments and simulations.