Functional Inequalities involving Nonlocal Operators on Complete Riemannian Manifolds and Their Applications to The Fractional Porous Medium Equation

TL;DR

This paper studies functional inequalities involving fractional Laplacians on Riemannian manifolds and applies these results to analyze the long-term behavior and well-posedness of the fractional porous medium equation.

Contribution

It derives new nonlocal functional inequalities on Riemannian manifolds and uses them to analyze solutions of the fractional porous medium equation.

Findings

Established nonlocal Sobolev-Poincaré, Nash, and logarithmic Sobolev inequalities.

Proved global well-posedness of the fractional porous medium equation on complete manifolds.

Analyzed asymptotic behavior of solutions to the fractional porous medium equation.

Abstract

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincar\'e, Nash, Super Poincar\'e and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, $\partial_t u +(-\Delta)^\sigma (|u|^{m-1}u )=0 $ with $m>0$ and $\sigma\in (0,1)$. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.