Continuous maximal regularity on singular manifolds and its applications

TL;DR

This paper develops a theory of continuous maximal regularity for linear differential operators on singular manifolds, with applications to various degenerate or singular parabolic equations.

Contribution

It introduces a new regularity framework for operators on singular manifolds and applies it to complex nonlinear PDEs with degeneracies.

Findings

Established maximal regularity results for singular manifold operators

Applied theory to Yamabe flow, porous medium, p-Laplacian, and thin film equations

Provided insights into boundary blow-up and waiting time phenomena

Abstract

In this article, we set up the continuous maximal regularity theory for a class of linear differential operators on manifolds with singularities. These operators exhibit degenerate or singular behaviors while approaching the singular ends. Particular examples of such operators include differential operators defined on domains, which degenerate fast enough toward the boundary. Applications of the theory established herein are shown to the Yamabe flow, the porous medium equation, the parabolic $p$-Laplacian equation and the thin film equation. Some comments about the boundary blow-up problem, and waiting time phenomena for singular or degenerate parabolic equations can also be found in this paper.