An alternative proof of logarithmically improved Beale-Kato-Majda type extension criteria for smooth solutions to the Navier-Stokes equations

We give an alternative proof of a logarithmically improved Beale-Kato-Majda type extension criterion for smooth solutions to the Navier-Stokes equations in the whole space, which was shown by Fan et al. (2011). By our method, we can also establish a similar criterion to the above in case of the half space, bounded domains and exterior domains.