"Irregularization" of Systems of Conservation Laws

TL;DR

This paper introduces 'irregularization' methods for systems of conservation laws that generate more singular defects like delta shocks, motivated by physical dislocation dynamics, contrasting traditional smoothing approaches.

Contribution

It proposes a novel irregularization scheme to produce stationary delta shock defects, expanding the understanding of defect behavior in conservation laws.

Findings

Irregularization schemes create more singular defects.

Stationary delta shocks can be achieved through these schemes.

Application to dislocation dynamics demonstrates physical relevance.

Abstract

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.