Local kinetic-energy balance for smooth incompressible flow

Statement

For a smooth solution of the incompressible Navier–Stokes equations, set $e=|u|^2/2$ . The local kinetic-energy balance is

\partial_t e+\nabla\cdot((e+p)u)=\nu\Delta e-\nu|\nabla u|^2+f\cdot u.

Here pressure and force are normalized by the constant density.

Derivation

Take the scalar product of the momentum equation with $u$ . The identities $u\cdot\partial_tu=\partial_te$ , $u\cdot(u\cdot\nabla)u=\nabla\cdot(eu)$ , $u\cdot\nabla p=\nabla\cdot(pu)$ , and $u\cdot\Delta u=\Delta e-|\nabla u|^2$ give the equation. The divergence identities use $\nabla\cdot u=0$ .

For a rough distributional solution, these multiplications require justification; this smooth equality does not automatically persist.