Well-posedness for fractional Navier-Stokes equations in critical spaces close to ${\dot{B}}_{\infty ,\infty }^{-(2\beta -1)}({\mathbb{R}}^n)$

Abstract

In this paper, we prove the well-posedness for the fractional Navier-Stokes equations in critical spaces $G_n^{-(2\beta -1)}({\mathbb{R}}^n)$ and $BM{O}^{-(2\beta -1)}({\mathbb{R}}^n).$ Both of them are close to the largest critical space ${\dot{B}}_{\infty ,\infty }^{-(2\beta -1)}({\mathbb{R}}^n).$ In $G_n^{-(2\beta -1)}({\mathbb{R}}^n),$ we establish the well-posedness based on a priori estimates for the fractional Navier-Stokes equations in Besov spaces. To obtain the well-posedness in $BM{O}^{-(2\beta -1)}({\mathbb{R}}^n),$ we find a relationship between $Q_{\alpha ;\infty }^{\beta ,-1}({\mathbb{R}}^n)$ and $BMO({\mathbb{R}}^n)$ by giving an equivalent characterization of $BM{O}^{-\zeta }({\mathbb{R}}^n).$

17 pages