Local energy inequality

For a weak velocity-pressure pair, write $e=|u|^2/2$ . Assume $\nabla u\in L^2_{\rm loc}$ and $e,eu,pu,f\cdot u$ are locally integrable. The local energy inequality is

\partial_t e+\operatorname{div}((e+p)u)-\nu\Delta e +\nu|\nabla u|^2\leq f\cdot u

in distributions. Explicitly, every nonnegative test function $\phi\in C_c^\infty(Q)$ satisfies

\nu\int_Q|\nabla u|^2\phi\leq\int_Q e(\partial_t\phi+\nu\Delta\phi)+(e+p)u\cdot\nabla\phi+f\cdot u\,\phi.

It permits extra energy loss relative to the smooth local balance.

Local and global statements

Tests have compact support inside space-time. Obtaining a global energy inequality requires cutoff limits and endpoint information; the two definitions are not interchangeable without those arguments.