ar X iv : 0 90 6 . 51 40 v 1 [ m at h . A P ] 2 8 Ju n 20 09 WELL - POSEDNESS FOR FRACTIONAL NAVIER - STOKES EQUATIONS IN CRITICAL SPACES

In this paper, we prove the well-posedness for the fractional NavierStokes equations in critical spaces G −(2β−1) n (R ) and BMO−(2β−1)(Rn). Both of them are close to the largest critical space Ḃ −(2β−1) ∞,∞ (R ). In G −(2β−1) n (R ), 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−(2β−1)(Rn), we find a relationship between Q α;∞ (R ) and BMO(R) by giving an equivalent characterization of BMO(R).