Weighted Lojasiewicz inequalities and regularity of harmonic map flow

At a finite-time singularity of harmonic map flow in the critical dimension, we show that a Lojasiewicz inequality between the quantities appearing in Struwe’s monotonicity formula implies continuity of the body map and the no-neck property for bubble-tree decompositions. We prove such an inequality when the target is $S^2,$ yielding both properties in this case.