Point singularities and nonuniqueness for the heat flow for harmonic maps

The simplest equation for the evolution of a director field is given by its corresponding heat flow. More complicated versions arise in the theories of micromagnetism and liquid crystals. In 3D there exist finite energy solutions with point singularities (also called defects in case of liquid crystals). It the paper an example of a new nonuniqueness phenomenon is discussed: having initially an equilibrium situation with one point singularity, a solution is constructed for which the singularity is moved instantaneously to another point. This suggests that there exists a considerable degree of freedom to prescribe the evolution of point singularities.