Non-uniqueness for continuous solutions to 1D hyperbolic systems

TL;DR

This paper demonstrates that for certain 1D hyperbolic systems, geometric conditions can lead to multiple continuous solutions, showing the Liu Entropy Condition alone does not ensure uniqueness, using convex integration.

Contribution

First application of convex integration to construct non-unique continuous solutions in one-dimensional hyperbolic systems.

Findings

Non-uniqueness arises under specific geometric conditions.

Liu Entropy Condition alone is insufficient for uniqueness.

Convex integration is used to construct multiple solutions.

Abstract

In this paper, we show that a geometrical condition on $2\times2$ systems of conservation laws leads to non-uniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone is insufficient to guarantee uniqueness, even within the mono-dimensional setting. We provide examples of systems where this pathology holds, even if they verify stability and uniqueness for small BV solutions. Our proof is based on the convex integration process. Notably, this result represents the first application of convex integration to construct non-unique continuous solutions in one dimension.