  The aim of this paper is to prove the strong convergence of the solutions to
a vector-BGK model under the diffusive scaling to the incompressible
Navier-Stokes equations on the two-dimensional torus. This result holds in any
interval of time $[0, T]$, with $T>0$. We also provide the global in time
uniform boundedness of the solutions to the approximating system. Our argument
is based on the use of local in time $H^s$-estimates for the model, established
in a previous work, combined with the $L^2$-relative entropy estimate and the
interpolation properties of the Sobolev spaces.
