Timestepping schemes for the 3d Navier-Stokes equations

It is well known that the (exact) solutions of the 3d Navier-Stokes equations remain bounded for all time if the initial data and the forcing are sufficiently small relative to the viscosity. They also remain bounded for a finite time for arbitrary initial data in L2. 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 uniformly bounded in H1 subject to essentially the same respective smallness conditions as the continuous system (on initial data and forcing or on the time of existence) provided the time step is small.