  In order to extend the blow-up criterion of solutions to the Euler equations,
Kozono and Taniuchi have proved a logarithmic Sobolev inequality by means of
isotropic (elliptic) $BMO$ norm. In this paper, we show a parabolic version of
the Kozono-Taniuchi inequality by means of anisotropic (parabolic) $BMO$ norm.
More precisely we give an upper bound for the $L^{\infty}$ norm of a function
in terms of its parabolic $BMO$ norm, up to a logarithmic correction involving
its norm in some Sobolev space. As an application, we also explain how to apply
this inequality in order to establish a long-time existence result for a class
of nonlinear parabolic problems.
