Hydrodynamic Singularities

The equations of hydrodynamics are nonlinear partial differential equations, so they include the possibility of forming singularities in finite time. This means that hydrodynamic fields become infinite or at least non-smooth at points, lines, or even fractal objects. This mathematical possibility is the price one has to pay for the enormous conceptual simplification a continuum theory brings for the description of a many-particle system. Near singularities, the microscopic structure re-emerges, as the flow changes over arbitrarily small distances. Eventually, the singularity is cut off by some microscopic length scale such as the distance between molecules. The most fundamental question is whether the microscopic structure becomes relevant for features of the flow much larger than the microscopic ones. If this is the case, the continuum description is no longer self-consistent, but has to be supplemented by microscopic information. There are some wellknown cases where singularities are artifacts of neglecting diffusive effects like viscosity in the hydrodynamic equations, and there is no more smoothing on small scales. Examples are the singularities widely believed to be produced by the three-dimensional Euler equation ([?]) or those of Hele-Shaw flows at zero surface tension ([?]). There does not seem to be a direct correspondence between the intricate structure of spatial singularities produced by these equations and real flows. For a realistic description of experiments, some tiny amount of viscosity or surface tension needs to be added. The reason hydrodynamic singularities have nevertheless attracted considerable attention in recent years is the observation that certain singularities have direct physical significance and are not a consequence of inadequacies of the equations. In particular, free-surface flows exhibit a rich variety of experimentally observable singularities, which are responsible for phenomena like the breakup of jets, coalescence of drops, and bubble entrainment.