We consider the stationary Navier-Stokes equations in $\mathbb{R}^n$ for $n\ge 3$. We show the existence and uniqueness of solutions in the homogeneous Triebel-Lizorkin space $\dot F^{-1+\frac{n}{p}}_{p,q}$ with $1 < p\leq n$ for small external forces in $\dot F^{-3+\frac{n}{p}}_{p,q}$. Our method is based on the boundedness of the Riesz transform, the para-product formula, and the embedding theorem in homogeneous Triebel-Lizorkin spaces. Moreover, it is proved that under some additional assumption on external forces, our solutions actually have more regularity.

中文翻译：

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尺度不变Triebel-Lizorkin空间中的平稳Navier-Stokes方程

我们考虑$ n \ ge 3 $的$ \ mathbb {R} ^ n $中的平稳Navier-Stokes方程。我们证明了齐次Triebel-Lizorkin空间$ \ dot F ^ {-1+ \ frac {n} {p}} _ {p，q} $中解的存在和唯一性，其中$ 1 <p \ leq n $ $ \ dot F ^ {-3+ \ frac {n} {p}} _ {p，q} $中的外力。我们的方法是基于齐兹Triebel-Lizorkin空间中Riesz变换的有界性，副产品公式以及嵌入定理。此外，事实证明，在对外力进行一些额外假设的情况下，我们的解实际上具有更多的规律性。