Instantaneous Boundary Tangency and Cusp Formation in Two-Dimensional Fluid Flow

We show that, for solutions of a model of two-dimensional, viscous, compressible fluid flow, curves which are initially transverse to the spatial boundary and across which the fluid density is discontinuous become tangent to the boundary instantaneously in time. We also show by similar techniques that, for the Euler equations of two-dimensional, inviscid, incompressible flow and for a large class of initial data corresponding to vortex patches with corners, these corner singularities become cusp-like instantaneously in time.