2D Constrained Navier-Stokes Equations

We study 2D Navier-Stokes equations with a constraint forcing the conservation of the energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\mathbb{R}^2$ and $\mathbb{T}^2$, by a fixed point argument. We also show that the solution of the constrained equation converges to the solution of the Euler equation as the viscosity $\nu$ vanishes.