Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow

We show a sharp version of the Sobolev inequality of the Beale-Kato-Majda and the Kozono-Taniuchi type in Lizorkin-Triebel space. As an application of this inequality, the regularity problem under the critical condition to the gradient flow of the harmonic map into a sphere is considered in the class L-2(0, T; BMO(R-n; S-m)), where BMO is the class of functions of bounded mean oscillations.