Knowlpedia

Definition

Relative kinetic energy

Half the squared L2 distance between two velocity fields.

For two velocities u,vu,v with v−u∈L2(Ω)v-u\in L^2(\Omega), their relative kinetic energy at unit density is

E(v∣u)=12∫Ω∣v−u∣2 dx.\mathcal E(v\mid u)=\frac12\int_\Omega|v-u|^2\,dx.

It is the of the difference field.

Comparison interpretation

It is nonnegative and vanishes exactly when u=vu=v almost everywhere. It is not the difference of their individual energies: 12∥v∥22−12∥u∥22\tfrac12\|v\|_2^2-\tfrac12\|u\|_2^2 also includes a cross term. A relative-energy estimate can turn a bound on one reference flow into stability or uniqueness of a competitor.