Nonuniqueness of the heat flow of director fields

Abstract

We give a new example of nonuniqueness of weak solutions of the initial-boundary-value problem for the heat flow of director fields in an infinitely long cylinder in $\mathbb R^{3}$. The example confirms the connection between nonuniqueness of axially symmetric solutions for the harmonic map heat flow and the occurrence of point singularities in the solutions. The result is compared with earlier nonuniqueness results. Traveling wave solutions are used as barrier functions.