On the Regularity of the Solution for Incompressible 3D Navier–Stokes Equation with Periodic Boundary Conditions

In this paper, we prove that the vorticity belongs to \(L^{\infty}(0,T;L^2(\Omega))\) for 3D incompressible Navier–Stokes equation with space-periodic boundary conditions, then the existence of a global smooth solution is obtained. Our approach is to construct a set of auxiliary systems to approximate the original system of vorticity equation.