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

TL;DR

This paper demonstrates that solutions of the Navier-Stokes equations on large periodic domains approximate solutions on the whole space, establishing a transfer of regularity from the entire space to periodic settings.

Contribution

It provides a rigorous analysis of how solutions on expanding periodic domains converge to whole-space solutions, illustrating a transfer of regularity for initial data with compact support.

Findings

Solutions on large periodic domains converge strongly to whole-space solutions.

Regularity of solutions transfers from the whole space to periodic domains as size increases.

Approximation holds for initial data with compact support in velocity or vorticity.

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.