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

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