A partial boundary regularity and a regularity criterion for Harmonic Heat flow

TL;DR

This paper advances the understanding of harmonic heat flow by establishing boundary regularity inequalities and a new regularity criterion, leading to partial and full domain regularity results under specific conditions.

Contribution

It introduces boundary energy inequalities and a boundary regularity criterion for harmonic heat flow, extending previous work with new boundary and domain regularity results.

Findings

Established boundary energy inequalities for harmonic heat flow.

Proved partial boundary regularity near the boundary.

Demonstrated full domain regularity under a one-sided condition.

Abstract

In my previous paper I have contrived a Ginzburg-Landau heat flow with a time-dependent parameter and by using it, I constructed a harmonic heat flow into spheres with a monotonical inequality and a reverse Poincar\'{e} inequality. This paper establishes these two energy inequalities near the boundary and then by making the best of them, we discuss a partial boundary regularity. In addition to it, we demonstrate a whole domain's regularity under "the one-sided condition." This has been proposed by S.Hildebrandt and K.-O.Widman.