Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

TL;DR

This paper establishes an energy identity for approximate Dirac-harmonic maps from surfaces with boundary during blow-up processes, with applications to heat flow blow-up analysis.

Contribution

It proves the energy identity for a class of approximate Dirac-harmonic maps near the boundary, extending understanding of blow-up behavior in these systems.

Findings

Energy identity holds during boundary blow-up.

Application to heat flow blow-up at infinite time.

Validates energy conservation in approximate solutions.

Abstract

For a sequence of coupled fields $\{(\phi_n,\psi_n)\}$ from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.