I'm trying to derive the weak formulation for the Navier-Stokes equations with boundaries imposed via Lagrange Multipliers. This technique is used by Urquiza 2014 for the Stokes equations. It's done by forming a minimization problem. From Caglar 2004, he's applied this method to the Navier-Stokes equations but has completely missed out how he got from equation (2.1) to (2.2) (although he says it's a straight forward substitution, you notice that the test function v is now multiplied by the constraint, either the unit normal, $n$, or unit tangents, $t_1$, $t_2$).
So I think what I have to do is to recast the Navier-Stokes formulation as an optimisation problem, then find it's stationary points, as Urquiza did.
However, I've been reading around and some sources says that no such formulation exists for the Navier-Stokes equations. I just want to confirm if this is true, if not, can someone nudge me in the right direction?
Many thanks!