  Studies on singular flows in which either the velocity fields or the
vorticity fields change dramatically on small regions are of considerable
interests in both the mathematical theory and applications. Important examples
of such flows include supersonic shock waves, boundary layers, and motions of
vortex sheets, whose studies pose many outstanding challenges in both
theoretical and numerical analysis. The aim of this talk is to discuss some of
the key issues in studying such flows and to present some recent progress.
First we deal with a supersonic flow past a perturbed cone, and prove the
global existence of a shock wave for the stationary supersonic gas flow past an
infinite curved and symmetric cone. For a general perturbed cone, a local
existence theory for both steady and unsteady is also established. We then
present a result on global existence and uniqueness of weak solutions to the
2-D Prandtl's system for unsteady boundary layers. Finally, we will discuss
some new results on the analysis of the vortex sheets motions which include the
existence of 2-D vortex sheets with reflection symmetry; and no energy
concentration for steady 3-D axisymmetric vortex sheets.
