The elusive singularity

In 1934 J. Leray (1) proved that Navier–Stokes equations—the partial differential equations used to describe viscous incompressible fluids—admit global weak solutions. The term global refers to the fact that there are no restrictions, neither in the size of the initial data nor in the length of time these solutions persist. The term weak is technical and refers to the smoothness of these solutions: they are not very smooth. Leray’s work did not preclude the possibility that, starting from smooth data, a singularity might form in finite time. Sixty-three years later we still don’t know. The spontaneous generation and subsequent evolution of singularities in smooth solutions of other well posed partial differential equations are somewhat better understood. For instance, discontinuities—shocks—arise spontaneously quite often in systems of hyperbolic equations (2). These singularities derive from the compressible and hyperbolic nature of the equations.

Do incompressible finite time singularities exist? Yes. When a drop of water falls from a faucet, the topology changes and the continuum mechanics description of the drop becomes singular. Mathematically, this is a free-boundary problem: the shape of the drop and the motion of the fluid are interdependent. Gravity or some other external force is the agent that produces the singularity. Much is known (3) about models describing the way a fluid breaks. In particular, it is believed that the formation of singularity is locally self-similar: near a place where the drop breaks the shape of the drop remains almost unchanged if one makes appropriate rescalings of spatial and temporal coordinates.

A singularity in an unforced inviscid incompressible fluid is more elusive: just as the viscous Navier–Stokes equations, the inviscid Euler equations still pose this basic challenge to mathematicians. Although an unforced finite time singularity in the Navier–Stokes equation is unphysical, finite time singularities in the Euler equations (4) are hoped to shed light on issues related to turbulence (5, 6). A paper in this issue of the Proceedings (7) considers the problem of singularity formation in an active scalar equation. This equation was proposed as a didactic model for unforced incompressible inviscid singularities (8) but has intrinsic interest as a geophysical model (9). The singularity corresponds to a sharp thermal front. Analysis and numerical experiment (10) suggested significant gradient amplification in the case of a hyperbolic saddle. Whether this is an infinite time singularity, as suggested in ref. 11, or a finite time singularity is not easy to decide on the basis of numerical data. The model, a partial differential equation with many affinities to the Euler equation, is simpler, in spite of nonlinearity and nonlocality. Nevertheless, the result of ref. 7 is that the formation of a singularity cannot be simple. It was known (10, 12) that the blow-up cannot happen without a certain degree of underlying geometrical complexity; for instance, the scalar cannot form a shock across a smooth front. The result of Cordoba (7) rules out a shock across a collapsing hyperbolic saddle, a specific scenario suggested by numerical simulations. Of added significance is the method of proof that suggests that perhaps a simple self-similar blow-up is not possible in this model.