论文标题
使用周期性的边界条件近似于$ \ mathbb {r}^3 $的Navier-Stokes方程和规律性的转移
Using periodic boundary conditions to approximate the Navier-Stokes equations on $\mathbb{R}^3$ and the transfer of regularity
论文作者
论文摘要
本文考虑了三维Navier的解决方案$u_α$ - 在周期域上stokes方程$q_α：=（ - α，α）^3 $作为域大小$α\ to \ infty $，并将它们与整个空间相同方程的解决方案进行比较。对于紧凑型的初始数据$u_α^0 \在h^1（q_α）$中，$u_α$的适当扩展收敛到$ {\ mathbb r}^3 $上的解决方案$ u $，在$ l^r（0，t;当$u_α^0 $是对应于固定的紧凑型涡度时的速度时，也同样也同样。 一个结果是，如果最初的紧凑型速度$ u_0 \ in H^1（{\ Mathbb r}^3）$或初始的紧凑型固定的涡度$ω_0\ in H^1（{\ Mathbb r}^3）$在$ [0，$ [\ Mathb r}^3）上$ [$ [\ Mathb r}^3）$ [0] r}^3 $，对于$ {q_α} $，对于$ {q_α} $，对于$α$，对于$ {q_α} $，也将存在一个平滑的解决方案，以$ [0，t^*] $足够大；这说明了从整个空间转移到周期性情况的“规律性转移”。
This paper considers solutions $u_α$ of the three-dimensional Navier--Stokes equations on the periodic domains $Q_α:=(-α,α)^3$ as the domain size $α\to\infty$, and compares them to solutions of the same equations on the whole space. For compactly-supported initial data $u_α^0\in H^1(Q_α)$, an appropriate extension of $u_α$ 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 $u_α^0$ 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 $ω_0\in H^1({\mathbb R}^3)$ 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 ${Q_α}$ for $α$ sufficiently large; this illustrates a `transfer of regularity' from the whole space to the periodic case.
