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