Transition Time for Weak Singularities of the Navier-Stokes Equations

This paper constructs a rigorous mathematical framework for investigating laminar-turbulent transition induced by weak singularities of incompressible Navier-Stokes (NS) equations. By integrating the energy identity of Leray weak solutions with the singularity criterion $\left\lVert \boldsymbol{u} \right\rVert_{H_0^1(\Omega)}\to0$, a closed analytical form of the laminar-turbulent transition characteristic time is derived. The theoretical scaling $t_{\text{trans}}\sim\nu/U^2$ (equivalent to $t_{\text{trans}}\sim t_c/\text{Re}$) is verified to be consistent with classical experimental observations in shear flows. This work reveals that laminar-turbulent transition is dominated by the local regularity collapse of Leray weak solutions rather than global viscous diffusion, and provides a novel theoretical interpretation for the onset of turbulence from the perspective of NS equation weak singularities.