On the geometric regularity conditions for the 3D Navier-Stokes equations

We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let upsilon and omega be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote {f}(+) = max{f, 0}, Q(T) = R-3 x (0, T). If {(upsilon x omega/vertical bar omega vertical bar). Lambda(beta)upsilon/vertical bar Lambda(beta)upsilon vertical bar}(+) is an element of L-x,t(gamma,alpha)(QT) with 3/gamma + 2/alpha <= 1 for some gamma > 3 and + 1 <= beta <= 2, then the local smooth solution upsilon of the Navier Stokes equations on (0, T) can be continued to (0,T + delta) for some delta > 0. We also prove localized version of a special case of this. Let v be a suitable weak solution to the Navier Stokes equations in a space time domain containing z(0) = (x(0), t(0)), let Q(z0,r) = B-x0,B-r x (t(0) - r(2), t(0)) be a parabolic cylinder in the domain. We show that if either {(upsilon x omega/vertical bar omega vertical bar).del x omega/vertical bar del omega vertical bar}(+) is an element of L-x,t(gamma,alpha)(Q(z0,r)) with 3/gamma + 2/alpha <= 1 or {(upsilon/vertical bar upsilon vertical bar x omega). del x omega/vertical bar del omega vertical bar}(+) is an element of L-x,t(gamma,alpha)(Q(z0,r)) with 3/gamma + 2/alpha <= 2 (gamma >= 2, alpha >= 2), then z(0) is a regular point for upsilon. This improves previous local regularity criteria for the suitable weak solutions. 2016 Elsevier Ltd. All rights reserved.