| We obtain a complete spatio-temporal localization of a class of analytic conditions on the vorticity which prevents singularity formation in the 3-D Navier-Stokes flows. The global-in-space conditions have been known since the mid 80s (Beale-Kato-Majda condition) and the mid 90s (Beirao DaVeiga). However, mainly due to the non-locality of the Biot-Savart law, the localization was an open problem. In 2000, Kozono and Taniuchi presented a refinement of the Beale-Kato-Majda condition to the L1 integrability of the BMO norm of the vorticity. A localization of this class is obtained in localized spaces via div-curl lemmas. In addition, a new scaling-invariant class of hybrid geometric-analytic regularity conditions in which the coherence of the vorticity direction serves as a weight in the Lp,q space-time integrability of the vorticity magnitude is presented.;At the end we provide a new, direct proof of spatial analyticity of the most general class of "mild solutions" to the 3-D NSE---the Koch-Tataru solutions emanating from small initial data in BMO -1. Unlike the already existing proofs, our proof is based on estimates on a suitable sequence of complexified approximations to the 3-D NSE. This provides an explicit estimate on the dynamics of the domain of spatial analyticity locally in time. |
