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It is well known that the solution of the 3d Navier--Stokes equations remainsbounded if the initial data and the forcing are sufficiently small relative tothe viscosity, and for a finite time given any bounded initial data. In thisarticle, we consider two temporal discretisations (semi-implicit and fullyimplicit) of the 3d Navier--Stokes equations in a periodic domain and provethat their solutions remain bounded in $H^1$ subject to essentially the samesmallness conditions (on initial data, forcing or time) as the continuoussystem and to suitable timestep restrictions.

Timestepping schemes for the 3d Navier-Stokes equations: small solutions and short times