The Navier-Stokes (NS) system of equations is a central paradigm of nonlinear partial differential equations describing nonintegrable dynamics. The mathematical analysis of the NS system invokes a priori estimates for the energy and enstrophy. The difficulty stemming from the vortex-tube stretching effect is explained. By replacing the convective nonlinear term by a random noise term, one can develop a statistical model of turbulence. The mathematical framework of such modeling is also reviewed.
