right-hand state travels down the 1-shock curve), and we ask that the shock strength increase in the sense of relative entropy (an $L^2$ requirement) as the right-hand state travels down the 1-shock curve. If we want to consider n-shocks, we ask for similar requirements on the n-shock family.

The intermediate wave families have far fewer requirements. The intermediate shock curves might not even be well-defined and characteristic speeds might cross.

In particular, the results in this article apply to both the isentropic Euler system and the full Euler system for a polytropic gas, viewing both systems in Eulerian coordinates.

We study solutions $\bar{u}$ which are piecewise-Lipschitz continuous in the space variable $x$. We study the stability and uniqueness of these solutions among a large class of weak solutions $u$ which are bounded, measurable, entropic for at least one strictly convex entropy, and verify a strong trace condition (weaker than $BV_{\text{loc}}$). We do not make small data assumptions. We require the piecewise-smooth $\bar{u}$ contain a single shock of extremal family. However, the rougher solutions $u$ which we compare to this solution $\bar{u}$ may have shocks of any type or family.

Previous results in the theory of stability and a-contraction have only 