"Irregularization" of Systems of Conservation Laws

We explore new ways of regulating defect behavior in systems of conservation laws. Contrary to usual regularization schemes (such as a vanishing viscosity limit), which attempt to control defects by making them smoother, our schemes result in defects which are \textit{more singular}, and we thus refer to such schemes as "irregularizations". In particular, we seek to produce \textit{delta shock} defects which satisfy a condition of \textit{stationarity}. We are motivated to pursue such exotic defects by a physical example arising from dislocation dynamics in materials physics, which we describe.