In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space $${\cal F}\dot {\cal N}_{p\left( \cdot \right),h\left( \cdot \right),q}^{s\left( \cdot \right)}\left( {{\mathbb{R}^3}} \right)$$ ℱ N ˙ p ( ⋅ ) , h ( ⋅ ) , q s ( ⋅ ) ( ℝ 3 ) with $$s\left( \cdot \right) = 4 - 2\alpha - {3 \over {p\left( \cdot \right)}}$$ s ( ⋅ ) = 4 − 2 α − 3 p ( ⋅ ) . We prove global well-posedness result with small initial data belonging to $${\cal F}\dot {\cal N}_{p\left( \cdot \right),h\left( \cdot \right),q}^{4 - 2\alpha - {3 \over {p\left( \cdot \right)}}}\left( {{\mathbb{R}^3}} \right)$$ ℱ N ˙ p ( ⋅ ) , h ( ⋅ ) , q 4 − 2 α − 3 p ( ⋅ ) ( ℝ 3 ) The result of this paper extends some recent work.

中文翻译：

---

变指数傅里叶-贝索夫-莫雷空间中分数阶 Navier-Stokes 方程的全局适定性

在本文中，我们研究了临界变量指数 Fourier-Besov-Morrey 空间中不可压缩分数 Navier-Stokes 方程的柯西问题 $${\cal F}\dot {\cal N}_{p\left( \cdot \right ),h\left( \cdot \right),q}^{s\left( \cdot \right)}\left( {{\mathbb{R}^3}} \right)$$ ℱ N ˙ p ( ⋅ ) , h ( ⋅ ) , qs ( ⋅ ) ( ℝ 3 ) 与 $$s\left( \cdot \right) = 4 - 2\alpha - {3 \over {p\left( \cdot \right)} }$$ s ( ⋅ ) = 4 − 2 α − 3 p ( ⋅ ) 。我们用属于 $${\cal F}\dot {\cal N}_{p\left( \cdot \right),h\left( \cdot \right),q 的小初始数据证明了全局适定性结果}^{4 - 2\alpha - {3 \over {p\left( \cdot \right)}}}\left( {{\mathbb{R}^3}} \right)$$ ℱ N ˙ p ( ⋅ ) , h ( ⋅ ) , q 4 − 2 α − 3 p ( ⋅ ) ( ℝ 3 ) 本文的结果扩展了最近的一些工作。