Kinetic energy of an incompressible velocity field

With constant density normalized to one, the kinetic energy of a velocity field on $\Omega$ at time $t$ is

E(t)=\frac12\int_\Omega|u(t,x)|^2\,dx =\frac12\|u(t,\cdot)\|_{L^2(\Omega)}^2.

It is finite exactly when the velocity belongs to spatial $L^2$ . For physical constant density $\rho$ , multiply the expression by $\rho$ .

Uniform energy bounds

Uniformly bounded kinetic energy specifies a bound uniform in time. This alone does not bound spatial derivatives or the pointwise magnitude of velocity. Energy identities and inequalities use the equation and appropriate boundary or decay assumptions in addition to this definition.

Energy estimates

See the global energy identity, energy inequality, and viscous dissipation.