This talk will describe two open problems. The first concerns the homogenous action of discrete dislocation pile-ups in a crystal lattice. The solution of the resulting large system of nonlinear algebraic equations can be approximated in terms of a dislocation density, but this density tends to infinity at the head of the pile-up.
The second concerns a continuum model for ion flow in the vicinity of a dust particle in a plasma. For sufficiently large ion velocity, the ion trajectories intersect each other but the nature of the reality singularities is unclear.