  We study the uniqueness, the continuity in $L^2$ and the large time decay for
the Leray solutions of the $3D$ incompressible Navier-Stokes equations with the
nonlinear exponential damping term $a (e^{b |u|^{\bf 2}}-1)u$, ($a,b>0$)
studied by the second author in \cite{J1}.
