This talk will discuss results on singularity formation for two different fluid flow problems,
and the physical significance of these results.

The first problem is the so-called Muskat problem, which describes the evolution
of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell. In contrast to the Hele-Shaw problem (the one phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem.

For the stable Muskat problem, in which the higher viscosity fluid expands into the lower-viscosity
fluid, it is shown that smooth solutions exist for all t>0, even if the initial data contains weak (e.g., curvature) singularities.

For the unstable problem, in which the higher viscosity fluid contracts, solutions are constructed that start
off smooth but develop a singularity in finite time.

The second example is the unsteady Prandtl equations for flow in an incompressible boundary layer.
A semi-analytic method for constructing singular solutions is suggested, and some preliminary results
toward this construction are presented.

This is joint work with Russ Caflisch and Sam Howison.