Weak Singularity of Navier-Stokes Equations Based on Energy Estimation in Sobolev Space

Based on Dou Huashu’s energy gradient theory, this paper focuses on the weak singularity of the incompressible Navier-Stokes (NS) equations in steady, fully developed flows. When the gradient of total mechanical energy is perpendicular to the streamline (i.e., $u_j\frac{\mathrm{\partial }E}{\mathrm{\partial }{x}_j}=0$), substituting this critical condition into the NS equations with no-slip boundary conditions leads to the viscous term $\nu \to 0$. To rigorously analyze the regularity of the solution, Sobolev space $H_0^1(\mathrm{\Omega })$ is introduced for energy estimation. The results show that the velocity field loses $H^1$-regularity, and the NS equations degenerate into Euler equations, which admit discontinuous weak solutions. Thus, the position where the mechanical energy gradient is perpendicular to the streamline becomes a weak singularity of the NS equations.