Point Singularities and Nonuniqueness for the Heat Flow for Harmonic Maps

Abstract

Abstract We consider weak solutions of the harmonic map flow between the three-dimensional unit ball B 3and the two-dimensional unit sphere, with as initial and boundary data. In this situation, we show the existence of infinitely many weak solutions. Indeed for any different from the origin we construct a weak solution whose singular set for all positive time is precisely Qand which satisfies a modified energy inequality.