The blow-up locus of heat flows for harmonic maps

Let M and N be tyro compact Riemannian manifolds. Let u(k)(x, t) be a seque nce of strong stationary weak heat flows from M x Rt to N with bounded ener gies. Assume that u(k) --> u weakly in H-1,H-2(M X R+, N) and that Sigma(t) is the blow-up set for a fixed t > 0. In this paper we first prove Sigma(t ) is an Hm-2-rectifiable set for almost all t is an element of R+. And then we prove two blow-up formulas for the blow-up set and the limiting map. Fr om the formulas, we can see that if the limiting map u is also a strong sta tionary weak heat flow, Sigma(t) is a distance solution of the (m - 2)-dime nsional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow -up set U(t<0)Sigma(t) x {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.