  In this work we consider the two dimensional instationary Navier-Stokes
equations with homogeneous Dirichlet/no-slip boundary conditions. We show error
estimates for the fully discrete problem, where a discontinuous Galerkin method
in time and inf-sup stable finite elements in space are used. Recently, best
approximation type error estimates for the Stokes problem in the
$L^\infty(I;L^2(\Omega))$, $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms
have been shown. The main result of the present work extends the error estimate
in the $L^\infty(I;L^2(\Omega))$ norm to the Navier-Stokes equations, by
pursuing an error splitting approach and an appropriate duality argument. In
order to discuss the stability of solutions to the discrete primal and dual
equations, a specially tailored discrete Gronwall lemma is presented. The
techniques developed towards showing the $L^\infty(I;L^2(\Omega))$ error
estimate, also allow us to show best approximation type error estimates in the
$L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms, which complement this
work.
