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This paper considers solutions $u_\alpha$ of the three-dimensionalNavier--Stokes equations on the periodic domains $Q_\alpha:=(-\alpha,\alpha)^3$as the domain size $\alpha\to\infty$, and compares them to solutions of thesame equations on the whole space. For compactly-supported initial data$u_\alpha^0\in H^1(Q_\alpha)$, an appropriate extension of $u_\alpha$ convergesto a solution $u$ of the equations on ${\mathbb R}^3$, strongly in$L^r(0,T;H^1({\mathbb R}^3))$, $r\in[1,\infty)$. The same also holds when$u_\alpha^0$ is the velocity corresponding to a fixed, compactly-supportedvorticity.  A consequence is that if an initial compactly-supported velocity $u_0\inH^1({\mathbb R}^3)$ or an initial compactly-supported vorticity $\omega_0\inH^1({\mathbb R}^3)$ gives rise to a smooth solution on $[0,T^*]$ for theequations posed on ${\mathbb R}^3$, a smooth solution will also exist on$[0,T^*]$ for the same initial data for the periodic problem posed on${Q_\alpha}$ for $\alpha$ sufficiently large; this illustrates a `transfer ofregularity' from the whole space to the periodic case.

Using periodic boundary conditions to approximate the Navier-Stokes equations on $\mathbb{R}^3$ and the transfer of regularity