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

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