**
Scaling Theory of Singularities
**

**Jens Eggers**

*Department of Physics,
Universitaet Essen,
45117 Essen,
Germany*

**Abstract: **
Owing to the nonlinear character of the equations of motion describing
fluid motion, sudden discontinuities in the pressure, the density,
or the surface shape of a fluid often occur. Examples are shock waves,
the separation of a fluid drop, bubbles rising in a viscous fluid,
or drops in strong electric fields. Singularities are the crucial events
in the evolution of a flow, and signal the emergence of new structures.
Physically, smaller and smaller structures are produced near the
singularity, so it lacks a characteristic length scale. Singular solutions
thus must be invariant under a change of scale, and have a simple
scaling form. Moreover, singularities turn out to be universal,
so the same structure is found in a broad variety of circumstances.
This talk explores the mathematics and the physical consequences of
these two fundamental properties.