for the system , with associated entropy flux $u\mapsto q(u;v)$. Furthermore, if $u$ is a weak solution to and entropic for $\eta$, then $u$ will also be entropic for $\eta(\cdot|v)$. This can be calculated directly from and – note that the map $u\mapsto\eta(u|v)$ is basically $\eta$ plus a linear term.

Moreover, by virtue of $\eta$ being *strictly* convex, the relative entropy is comparable to the $L^2$ distance, in the following sense:

\[entropy\_relative\_L2\_control\_system\] For any fixed compact set $V\subset\mathcal{V}$, there exists $c^*,c^{**}>0$ such that for all $u,v\in V$, $$\begin{aligned}
c^*\abs{a-b}^2\leq \eta(u|v)\leq c^{**}\abs{a-b}^2.\end{aligned}$$ The constants $c^*,c^{**}$ depend on $V$ and bounds on the second derivative of $\eta$.

This lemma follows from Taylor’s theorem; for a proof see [@Leger2011; @VASSEUR2008323].

Given a Lipschitz solution $\bar{u}$ to , and a weak, entropic solution $u$, the method of relative entropy gives estimates on the growth in time of the quantity $$\begin{aligned}
\norm{\bar{u}(\cdot,t)-u(\cdot,t)}_{L^2(\mathbb{R})}\end{aligned}$$ by studying the time derivative $\partial_t\int\eta(u|\bar{u})\,dx$ and using the entropy inequality . By , we get $L^2$-type stability estimates.

Introducing a discontinuity into $\bar{u}$ causes difficulties in the method of relative entropy. In particular, simple examples for the scalar conservation laws show that a discontinuity in $\bar{u}$ prevents stability between $\bar{u}$ and $u$ 