J. math. fluid mech. 10 (2008) 106–125 1422-6928/08/010106-20 c  2006 Birkh¨ auser Verlag, Basel DOI 10.1007/s00021-006-0220-z Journal of Mathematical Fluid Mechanics New Sufficient Conditions of Local Regularity for Solutions to the Navier–Stokes Equations A. Mahalov, B. Nicolaenko and G. Seregin Communicated by V. A. Solonnikov Abstract. New sufficient conditions of local regularity for suitable weak solutions to the non- stationary three-dimensional Navier–Stokes equations are proved. They contain the celebrated Caffarelli–Kohn–Nirenberg theorem as a particular case. Mathematics Subject Classification (2000). 35K, 76D. Keywords. Navier–Stokes equations, suitable weak solutions, local regularity. 1. Introduction The aim of the paper is to present a unified approach to derivation of conditions ensuring local regularity of weak solutions to the non-stationary three-dimensional Navier–Stokes equations. They contain the famous Caffarelli–Kohn–Nirenberg theorem, see [1], as a particular case. Their result is formulated in terms of the scaled integral of the velocity gradient. It is worthy to notice that this integral is invariant with respect to the natural scaling for the Navier–Stokes equations. We show that the Caffarelli–Kohn–Nirenberg statement remains to be valid if the gradient of the velocity field is replaced with symmetric or skew part of it. Although the problem is interesting itself, for us, in the first place, it is motivated by the theory of rotating fluids. To explain the reason for that, let us consider a container filled up with a viscous liquid. The container is rotated and we may describe what is going on in it using either absolute or rotating coordinate system. Do local regularity conditions exist that have the same form in both coordinate systems? It is very important if one wishes to figure out what happens locally in very fast rotating fluids. For example, the symmetric part of the gradient gives us such kind of conditions. We hope that our work is the first step in understanding local properties of rotating fluids. Now, let us state main results of the paper. To this end, we consider the