Examples of identities and inequalities for the nonlinear term in the Navier–Stokes equation

In this paper, several examples of novel equations and inequalities for various forms of the non-linear term in the Navier–Stokes equations (NSE) are provided. The NSE are formulated from the conservation of linear momentum and mass conservation. However, they are also known to conserve energy, angular momentum, enstrophy in 2D, helicity in 3D, among other important physical quantities (Gresho and Sani, 1998) [1]. Depending on the desired quantity of interest, there are various representations of nonlinear term $u\cdot \nabla u$ (e.g. convective, skew symmetric, rotational etc.) that can be implemented.