LOGARITHMICALLY IMPROVED REGULARITY CRITERIA FOR THE NAVIER-STOKES EQUATIONS IN HOMOGENEOUS BESOV SPACES

We investigate a logarithmically improved regularity criteria in terms of the velocity, or the vorticity, for the Navier-Stokes equations in homogeneous Besov spaces. More precisely, we prove that if the weak solution u satisfies either integral(T)(0) parallel to u(t)parallel to(2/1-alpha)((B)over dot infinity,infinity-alpha)/1 + log(+) parallel to u(t)parallel to((H)over dots0) dt < infinity, or integral(T)(0) parallel to w(t)parallel to(2/2-alpha)((B)over dot infinity,infinity-alpha)/1 + log(+) parallel to w(t)parallel to((H)over dots0) dt < infinity, where w = rot u, then u is regular on (0, T]. Our conclusions improve some results by Fan et al. [5].