New functional inequalities with applications to the arctan-fast diffusion equation

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.