  In this paper, we prove a couple of new nonlinear functional inequalities of
Sobolev type akin to the logarithmic Sobolev inequality. In particular, one of
the inequalities reads $$ \int_{\mathbb{S}^1}\arctan\left(\frac{\partial_x
u}{u}\right)\partial_xu \,dx\geq
\arctan\left(\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}\right)\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}.
$$ Then, these inequalities are used in the study of the nonlinear
\emph{arctan}-fast diffusion equation $$ \partial_t
u-\partial_x\arctan\left(\frac{\partial_x u}{u}\right)=0. $$ For this highly
nonlinear PDE we establish a number of well-posedness results and qualitative
properties.
