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

Abstract

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 $\alpha\to\infty$, 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 $L^r(0,T;H^1({\mathbb R}^3))$, $r\in[1,\infty)$. 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 $u_0\in H^1({\mathbb R}^3)$ or an initial compactly-supported vorticity $\omega_0\in H^1({\mathbb R}^3)$ gives rise to a smooth solution on $[0,T^*]$ for the equations 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 of regularity' from the whole space to the periodic case.