Property (b), which says that the 1-shock strength grows along the 1-shock curve $S^1_{u_L}$ when measured via the pseudo-distance of the relative entropy (recall that the map $(u,v)\mapsto\eta(u|v)$ measures $L^2$-distance somehow – see ). This growth condition arises naturally in the study of admissibility criteria for systems of conservation laws. In particular, Property (b) ensures that Liu admissible shocks are entropic for the entropy $\eta$ even for moderate-to-strong shocks (see [@MR1600904; @MR0093653; @MR2053765]).

    In [@MR3338447], Barker, Freistühler, and Zumbrun show that stability and in particular contraction fails to hold for the full Euler system if we replace Property (b) with $$\begin{aligned}
    \frac{\mbox{d}}{\mbox{d}s}\eta(S_u^1(s))>0,\hspace{.2in} s>0.\end{aligned}$$ This shows that it is better to measure shock strength using the relative entropy rather than the entropy itself.

-   Recall the famous Lax E-condition for an i-shock $(u_L,u_R,\sigma)$, $$\begin{aligned}
    \lambda_i(u_R)\leq\sigma\leq\lambda_i(u_L).\end{aligned}$$ The hypothesis $(\mathcal{H}2)$ is implied by the first half of the Lax E-condition along with the hyperbolicity of the system . In addition, we do not allow for right 1-contact discontinuities.

-   The hypothesis $(\mathcal{H}3)$ is a statement about the well-separation of the 1-shocks from all other Rankine-Hugoniot discontinuities entropic for $\eta$; the 1-shocks do not interfere with any other shocks. In particular, $(\mathcal{H}3)$ will hold 