Global existence and blow-up for harmonic map heat flow

Abstract

For degree-one equivariant maps on bounded domains, the question of finite-time blow-up vs. global existence of solutions to the harmonic map heat flow has been well studied. In this paper we study the Cauchy problem for degree- m equivariant harmonic map heat flow from ( 2 + 1 ) -dimensional space–time into the 2-sphere with initial energy close to the energy of harmonic maps. It is proved that solutions are globally smooth for m ⩾ 4 , whereas for m = 1 , we show that finite-time singularities can form for this class of data.