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

TL;DR

This paper proves the well-posedness of fractional Navier-Stokes equations in critical function spaces close to the largest known critical space, using a priori estimates and characterizations of related function spaces.

Contribution

It establishes well-posedness in new critical spaces for fractional Navier-Stokes equations, extending previous results to spaces near the largest critical space.

Findings

Well-posedness in $G^{-(2eta-1)}_{n}( R^{n})$ established.

Well-posedness in $BMO^{-(2eta-1)}( R^{n})$ demonstrated.

Relationship between $Q_{eta; finite}^{eta,-1}( R^{n})$ and $BMO( R^{n})$ clarified.

Abstract

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 $BMO^{-\zeta}(\mathbb{R}^{n}).$