 Minimum-dissipation theorem for Stokes flow

Among all incompressible velocity fields with the same prescribed boundary velocity and far-field behavior, the Stokes flow minimizes viscous dissipation. If $\mathbf{v}=\mathbf{u}+\mathbf{w}$, where $\mathbf{u}$ is the Stokes solution and $\mathbf{w}$ has homogeneous boundary data, integration by parts and the Stokes equation eliminate the cross term, leaving