  We describe a time-dependent functional involving the relative entropy and
the $\dot{H}^1$ seminorm, which decreases along solutions to the spatially
homogeneous Landau equation with Coulomb potential. The study of this monotone
functionial sheds light on the competition between the dissipation and the
nonlinearity for this equation. It enables to obtain new results concerning
regularity/blowup issues for the Landau equation with Coulomb potential.
