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

TL;DR

The paper introduces nonlinear Sobolev inequalities of arctan type on ${\mathbb S^1}$ and employs them to study the one-dimensional arctan-fast diffusion equation ${\partial_t u - \partial_x \arctan\left(\frac{\partial_x u}{u}\right)=0}$. It proves two main arctan-Sobolev inequalities, establishes local and global well-posedness for positive initial data in ${H^3({\mathbb S^1})}$, and derives entropy and energy balances together with a Lyapunov framework. A θ-formulation with ${\tan(\theta)=\frac{\partial_x u}{u}}$ yields bounded slope, a dissipative Lyapunov functional, and regularity results such as ${u \in L^{\infty}(0,T;W^{1,\infty})} \cap L^2(0,T;H^2)$. The paper also proves existence in Wiener spaces via a smallness condition in ${A^1({\mathbb S^1})}$ and a Galerkin approach, enabling results beyond smooth data. These contributions provide rigorous tools for nonlinear diffusion with arctangent nonlinearity and illuminate the long-time behavior of solutions in both classical and Wiener settings.

Abstract

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.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Existence through Lyapunov functional
• Theorem 1.6: Existence in Wiener spaces