  It is well known that the solution of the 3d Navier--Stokes equations remains
bounded if the initial data and the forcing are sufficiently small relative to
the viscosity, and for a finite time given any bounded initial data. In this
article, we consider two temporal discretisations (semi-implicit and fully
implicit) of the 3d Navier--Stokes equations in a periodic domain and prove
that their solutions remain bounded in $H^1$ subject to essentially the same
smallness conditions (on initial data, forcing or time) as the continuous
system and to suitable timestep restrictions.
