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We are concerned with the problem,originated from Seregin [18,19,20], whatare minimal sufficiently conditions for the regularity of suitable weaksolutions to the 3d Naiver-Stokes equations. We prove some interior regularitycriteria, in terms of either one component of the velocity with sufficientlysmall local scaled norm and the rest part with bounded local scaled norm, orhorizontal part of the vorticity with sufficiently small local scaled norm andthe vertical part with bounded local scaled norm. It is also shown that onlythe smallness on the local scaled $L^{2}$ norm of horizontal gradient withoutany other condition on the vertical gradient can still ensure the regularity ofsuitable weak solutions. All these conclusions improve pervious results on thelocal scaled norm type regularity conditions.

Anisotropic regularity conditions for the suitable weak solutions to the 3d Navier-Stokes equations