The best theory so far is the Bressan, Crasta, and Piccoli $L^1$ theory [@MR1686652] for uniqueness in the class of solutions with small total variation. It would be interesting however to study the uniqueness of these solutions amongst a larger class. For example, existence of solutions with large data is known for the $2\times2$ Euler system – but the uniqueness theory for such solutions with large data lags behind.

The situation for the hyperbolic conservation laws in multiple space dimensions is even more dire – there is non-uniqueness for entropic solutions to incompressible and compressible Euler by virtue of the many highly oscillatory solutions created via convex integration or related techniques. For incompressible Euler, see two papers by De Lellis and Székelyhidi [@MR2600877; @MR2564474]. For compressible Euler, see [@MR3352460; @MR3269641; @MR3744380].

However, there is still the possibility of pushing forward the theory of *uniqueness* for hyperbolic systems of conservation laws in one space dimension. The current paper is a step in that direction – utilizing the $L^2$-type relative entropy method and the constantly evolving theory of shifts.

In this article, we use the method of relative entropy, the theory of shifts and a-contraction. These theories are not perturbative. They enable 