is similar to the growth condition (\[minimality\]) in Theorem \[minimality\_theorem\]. In Chapter 9.5 of [@Da] it is proved:

[@Da] A weak solution $u$ of (\[scl\]) with initial data (\[riemann\]) satisfies the entropy rate admissibility criterion for an entropy-entropy flux pair $(\eta,\xi)$ if and only if $u$ satisfies the E-condition (\[e\_condition\]).

Again by (\[entropy\_rate\]) and (\[entropy\_rate2\]) one sees, that this Theorem establishes, similar as in Theorem \[minimality\_theorem\], a connection between growth rate of the entropy production (\[entropy\_production\]) and entropy admissibility conditions (\[e\_condition\]) and (\[e\_inequality\]). In Chapter 9.5 there is also an extension of this theorem in the case of strictly hyperbolic systems.

Another results relating an optimality criterion to entropic solution is given by A. Poliakovsky in [@Po]. For $u:\mathbb{R}^n\times[0,T]\to\mathbb{R}^k$ he considers a family of energy functionals $$\label{energy1}
I_{\varepsilon, f}(u)=\frac12\int_0^T\int_{\mathbb{R}^n}\left(\varepsilon|\nabla_x u|^2+\frac{1}{\varepsilon}|\nabla_x H|^2\right)\,dx\,dt+\frac12\int_{\mathbb{R}^n}|u(x,T)|^2\,dx\,$$ where $$\Delta_x H_u=\partial_t u+\operatorname{div}_x f(u)\,.$$ Under certain assumptions on the flux $f$ he shows, that there exists a minimizer to $$\inf \left\{ I_{\varepsilon,f}(u):~u(x,0)=u_0(x)\right\}$$ and this minimizer satisfies $$\left.\begin{array}{rcll}
\partial_t u+\operatorname{div}_x f(u)&=&\varepsilon\Delta_x H_u&\forall(x,t)\in\mathbb{R}^n\times(0,T)\,,\\
u(x,0)&=&u_0(x)&\forall x\in\mathbb{R}^n\,.
\end{array}\right\}$$ In the particular case $k=1$, he calculates the $\Gamma$-limit of (\[energy1\]) as $\varepsilon\to0^+$ and finds an alternative variational formulation of the admissibility criterion for the particular solutions to the scalar conservation laws that can 