Uniformly bounded kinetic energy

A velocity $u(t,x)$ has uniformly bounded kinetic energy on a time interval $I$ if

\operatorname*{ess\,sup}_{t\in I}\frac12\int_\Omega|u(t,x)|^2\,dx<\infty.

The integral is kinetic energy at unit density. This is the condition $u\in L^\infty_t(I;L^2_x(\Omega))$ .

Time and space distinctions

The essential supremum ignores a null set of times. For an $L^2$ -continuous representative, it agrees with the ordinary supremum on a compact interval. Finite energy separately at each time is weaker than a uniform bound. Neither condition controls the maximum velocity or its spatial derivatives without additional estimates.