us to get results without small data limitations. Further, by the nature of these theories, we only use a single entropy condition.

We present our main and most important theorem regarding $L^2$-type stability and uniqueness results. The hypotheses $(\mathcal{H})$ and $(\mathcal{H})^*$ in the theorem depend only on the hyperbolic part of the system and the fixed piecewise-smooth solution $\bar{u}$. The hypotheses are related to conditions on 1-shocks and n-shocks and in particular are satisfied by the isentropic Euler and full Euler systems. These hypotheses are explained in detail in .

\[local\_stability\_systems\]

Fix $R,T>0$.

Fix $i\in\{1,n\}$. Assume that $u,\bar{u}\in L^\infty(\mathbb{R}\times[0,T))$. If $\bar{u}$ contains a 1-shock, assume the hypotheses $(\mathcal{H})$ hold. Likewise, if $\bar{u}$ contains an n-shock, assume the hypotheses $(\mathcal{H})^*$ hold. Assume that $u$ and $\bar{u}$ are entropic for the entropy $\eta\in C^3(\mathbb{R}^n)$. Assume that $\bar{u}$ is Lipschitz continuous on $\{(x,t)\in\mathbb{R}\times[0,T) | x<s(t)\}$ and on $\{(x,t)\in\mathbb{R}\times[0,T) | x>s(t)\}$, where $s:[0,T)\to\mathbb{R}$ is a Lipschitz function . Assume also that $u$ verifies the strong trace property (). Assume also that there exists $\rho>0$ such that for all $t\in[0,T)$ $$\begin{aligned}
\label{gap_local_case}
\abs{\bar{u}(s(t)+,t)-\bar{u}(s(t)-,t)}>\rho.\end{aligned}$$

Then there exists a Lipschitz continuous function $X:[0,T)\to\mathbb{R}$ with $X(0)=0$ and constants $\mu_1,\mu_2,r>0$ such that,

$$\begin{aligned}
\label{main_local_stability_result}
\int\limits_{-R+s(0)}^{R+s(0)}\abs{u(x,t_0)-\bar{u}(x+X(t_0),t_0)}^2\,dx\leq \mu_2 e^{\mu_1 t_0}\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx ,\end{aligned}$$

