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 more singular, and we thus refer to such schemes as “irregularizations”. In particular, we seek to produce delta shock defects which satisfy a condition of stationarity. We are motivated to pursue such exotic defects by a physical example arising from dislocation dynamics in materials physics, which we describe.
Highlights
A remarkable feature of systems of conservation laws is the tendency of smooth initial data to give way to non-smooth defects after only finite time
In the second section we describe delta shocks and discuss various ways in which they can arise in systems of conservation laws, and we introduce the condition of stationarity and the irregularization mentioned above
In the former case, the partial differential equation (PDE) is modified by the addition of a small term that serves to smooth out the defects; for the Hopf equation, such a modification looks like ut +
Summary
A remarkable feature of systems of conservation laws is the tendency of smooth initial data to give way to non-smooth defects after only finite time. Some more exotic systems feature different classes of defects which require other types of regularization Fig. 1; for example, superfluids are known (Meppelink et al 2009) to exhibit dispersive shock waves which are described by a very small dispersive term. Still another type of defect (which is a primary subject of this paper) is a delta shock, which consists of a delta function-like spike, possibly riding atop a regular (viscous) shock. In the second section we describe delta shocks and discuss various
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