A boundary partial regularity and a regularity criterion for New Harmonic Heat flows

TL;DR

This paper develops boundary partial regularity results and a regularity criterion for harmonic heat flows, utilizing energy inequalities and boundary conditions to improve understanding of flow regularity near boundaries.

Contribution

It introduces new boundary energy inequalities and two sufficient conditions for the regularity of harmonic heat flows, advancing boundary regularity theory.

Findings

Established boundary monotonicity inequalities.

Derived a reverse Poincare inequality at boundary points.

Proposed two regularity criteria based on boundary energy smallness and one-sided conditions.

Abstract

In my previaou paper of K. Horihata, we have proposed a Ginzburg-Landau system with a time-dependent parameter and then passing to the limit we have constructed a harmonic heat flow into spheres. Thanks to this scheme, we establish a few energy inequalities of our flow: (i) monotonical inequalities and (ii) a reverse Poincare inequality at any boundary point. These inequalities (i) and (ii) derive the smaller estimates on the set on noncontinuouspoints for pur flow contrast to the former results. We refer to them by Y. Chen and Y. Chen, J. Li, F. H. Lin. Next we introduce two sufficient condition for the whole domain's regularity for it; The one is a boundary energy smallness and the another is an one-sided condition proposed by S. Hildebrandt and K. O. Widman.