Singularities in Fluid Dynamics and their Resolution

Three types of singularity that can arise in fluid dynamical problems will be distinguished and discussed. These are: (i) singularities driven by boundary motion in conjunction with viscosity (e.g. corner singularities, or the Euler-disc finite-time singularity); (ii) free-surface (cusp) singularities associated with surface-tension and viscosity; (iii) interior point singularities of vorticity associated with intense vortex stretching. The singularities of types (i) and (ii) are now well known, and mechanisms by which the singularities may be resolved are clear. The question of existence of singularities of type (iii) is still open; current evidence for and against will be discussed.