Shadow wave solutions for a scalar two-flux conservation law with Rankine-Hugoniot deficit

TL;DR

This paper introduces shadow wave solutions for scalar conservation laws with flux discontinuities, providing a novel approach to handle cases lacking classical Rankine-Hugoniot solutions by using unbounded, delta-supported solutions.

Contribution

The paper develops a new shadow wave framework to solve scalar conservation laws with flux discontinuities where classical solutions do not exist.

Findings

Shadow wave solutions effectively model flux discontinuities.

The approach captures unbounded and delta-supported behaviors.

Provides a new method for non-classical shock wave analysis.

Abstract

The paper deals with scalar conservation laws having a flux discontinuity at $x=0$ without a weak solution that satisfies the classical Rankine--Hugoniot jump condition at $x=0$. We are using unbounded solutions in the form of shadow waves supported by the origin for solving that problem. The shadow waves are nets of piecewise constant functions approximating a shock wave with added a delta function and sometimes another unbounded part.