Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

improved regularity of harmonic map flows with hölder continuous energy ∗

For a smooth harmonic map flow u : M× [0, T ) → N with blow-up as t ↑ T , it has been asked ([6], [5], [7]) whether the weak limit u(T ) : M→ N is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function E(u(·)) : [0, T ) → R, being weakly positive, smooth and weakly decreasing, has a continuous extension to [0, T ]. Here we show that if this exten...

On p-Harmonic Map Heat Flows for 1<=p<∞ and Their Finite Element Approximations

Motivated by emerging applications from imaging processing, the heat flow of a generalized p-harmonic map into spheres is studied for the whole spectrum, 1 ≤ p <∞, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a BV -solution concept is introduced and t...

Rigidity in the Harmonic Map Heat Flow

We establish various uniformity properties of the harmonic map heat ow, including uniform convergence in L 2 exponentially as t ! 1, and uniqueness of the positions of bubbles at innnite time. Our hypotheses are that the ow is between 2-spheres, and that the limit map and any bubbles share the same orientation.

Notes on the regularity of harmonic map systems

In this note, we provide an alternative proof of C-regularity of continuous weak solutions to the system of harmonic map or heat flow of harmonic maps by Riesz potential estimates between Morrey spaces.

Existence and Partial Regularity Results for the Heat Flow for Harmonic Maps