Regularity estimates for the p-Sobolev flow

TL;DR

This paper investigates the p-Sobolev flow, a doubly nonlinear parabolic equation related to Sobolev inequalities, establishing key estimates and regularity results crucial for understanding long-term behavior and limits.

Contribution

It provides the first a priori estimates and regularity results for the p-Sobolev flow, extending classical results to a broader nonlinear setting.

Findings

Established a priori estimates for the p-Sobolev flow

Proved regularity results necessary for analyzing long-term behavior

Set the stage for classification of flow limits as time approaches infinity

Abstract

We study doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which includes the classical Yamabe flow on a bounded domain in Euclidean space in the special case p=2. In this article we establish a priori estimates and regularity results for the $p$-Sobolev type flow, which are necessary for further analysis and classification of limits as time tends to infinity.