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@ 2009-07-01 08:02:00
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Well-posedness for fractional Navier-Stokes equations in critical spaces close to $\dot{B}^{-(2\beta

Well-posedness for fractional Navier-Stokes equations in critical spaces close to $\dot{B}^{-(2\beta-1)}_{\infty,\infty}(\mathbb{R}^{n})$. (arXiv:0906.5140v1 [math.AP])

Authors: Zhichun Zhai

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


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