We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{\nabla }^j{u(t,x)|=O(|x|}^{1-n-j})$ and $|{\nabla }^j{p(t,x)|=O(|x|}^{-n-j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.

中文翻译：

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Navier–Stokes方程的时间周期解的逐点衰减率

我们研究了整个空间中具有Navier-Stokes方程的逐点衰减特性的时间周期解的存在。我们表明，如果在适当的意义上，时间周期外力足够小，则存在一个时间周期解$\{ü，p\}$ Navier–Stokes方程的等式 $|{\nabla }^{Ĵ}{ü（Ť，X）|=Ø（|X|}^{\mathrm{1个}-ñ-Ĵ}）$ 和 $|{\nabla }^{Ĵ}{p（Ť，X）|=Ø（|X|}^{-ñ-Ĵ}）（Ĵ=0，\mathrm{1个}，\dots ）$ 统一在 $Ť\in \mathbb{[R}$ 如 $|X|\to \infty$。我们的解的衰减比时间周期Stokes基本解更快，并且还描述了其高阶空间导数的更快衰减。