# Optimal jump set in hyperbolic conservation laws

## TL;DR

This paper investigates the properties of entropy solutions to hyperbolic conservation laws, revealing that their jump sets can be dense and not closed, which challenges previous assumptions about their structure.

## Contribution

It introduces the first examples of entropy solutions with dense, non-closed jump sets for scalar and hyperbolic systems, using two different analytical approaches.

## Key findings

- Existence of entropy solutions with dense jump sets
- Explicit solutions for scalar conservation laws with non-closed jump sets
- Extension of results to hyperbolic systems

## Abstract

This paper deals with some qualitative properties of entropy solutions to hyperbolic conservation laws. In [11] the jump set of entropy solution to conservation laws has been introduced. We find an entropy solution to scalar conservation laws for which the jump set is not closed, in particular, it is dense in a space-time domain. In the later part of this article, we obtain a similar result for the hyperbolic system. We give two different approaches for scalar conservation laws and hyperbolic system to obtain the results. For the scalar case, obtained solutions are more explicitly calculated.