Blow-up rate estimates for weak solutions of the Navier-Stokes equations

The interior regularity problem for the Leray weak solutions a of the Navier-Stokes equations in a domain Omega subset of R-n with n greater than or equal to 3 is investigated. It is shown that a is regular in a neighbourhood of a point (x(0), t(0)) is an element of Omega x (0, T) if there exist constants 0 less than or equal to theta < 1 and small epsilon > 0 such that lim(k --> infinity) (Q1/K(X0,T0)) ess sup \t - t(0)\ (theta /2)\x-x(0)\ (1-theta)\u(x,t)\ < epsilon with Q(1/k)(x(0), t(0)) = {x is an element of R-n;\x - x(0)\ < 1/k} x (t(0) - 1/k(2),t(0) + 1/k(2)). If (x(0), t(0)) is an irregular point of u, there exists a sequence of non-zero measure sets E-ki subset of Q(1/ki)(x(0), t(0)) for i = 1,2..... such that the blow-up rate estimate \u(x, t)\ greater than or equal to epsilon \t - t(0)\ (-theta /2)\x - x(0)\ (-1+theta), (x, t) is an element of E-ki holds.