Selected topics of local regularity theory for navier-stokes equations

ℝ+ = {t ε ℝ: t > 0}, ℝ- ={t ε ℝ: t < 0}; ℝ+d = {x =(x′, xd) : ′ = (xi), i = 1,2,...,d 1 xd > 0};B(x, r) is the ball in of radius r centered at the point, B(r) = B(0, r), B = B(1) x > is a half ball, Q(z, r) = B(x, r) ×]t - r 2, t[ is the parabolic ball in of radius r centered at the point, Q(r) = Q(0, r), Q = Q(1) and are the usual Lebesgue and Sobolev spaces, respectively and are parabolic Sobolev spaces;; is the closure of the set in the space, is the closure of the same set with respect to the Dirichlet integral; BMO is the space of functions having bounded mean oscillation; © 2013 Springer-Verlag Berlin Heidelberg.