Singularities and Similarities in Interface Flows

Abstract

The onset of singularities in systems of nonlinear partial differential equations is an important issue in fields ranging from general relativity [27], to thermodynamic phase transitions [10], to fluid dynamics [13]. The development of a mathematical singularity, when some quantity associated with the PDE “blows up,” reflects the creation of a new structure in the physical system which in turn forces the mathematical formulation to change. Whether or not such singularities are possible for a given system can be a difficult question. A famous problem from the theory of homogeneous incompressible fluids is the question of finite time singularity development in the three-dimensional Navier-Stokes equation: It is unknown if an initially smooth solution can develop a finite time singularity in which the vorticity becomes unbounded [23]. To date, no rigorous proof or counterexample exists; neither numerical nor physical experiments have produced definitive answers [22, 25]. When a particular system allows finite time singularities, many related questions become relevant. For example, do all singularities have universal characteristics, or are there many possible behaviors? Which quantities are unbounded at the singular time?