Using periodic boundary conditions to approximate the Navier–Stokes equations on R3 and the transfer of regularity

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