by a weak solution of (\[scl\]), i.e. $$\mathcal{W}:=\left \{ 
\begin{array}{c}
m(x,t,a)\in \mathcal{M}_{loc} ~\text{s.t.}~m(x,t,a)=\partial_{t}(u\wedge a)+\partial_{x}f(u\wedge a),\\
\text{where}~u\in L^{\infty}~\text{is a weak sol. of (\ref{scl}).}\end{array}\right\}$$ Our main result in the present work is the following.

\[minimality\_theorem\] Let $f\in C^{2}(\mathbb{R})$ such that $f''\geq c>0$ and $$\label{infinity_behaviour}
 \lim_{|x|\rightarrow\infty}f(x)=\infty\,.$$ Moreover let $u_{0}\in L^{\infty}(\mathbb{R})$ be compactly supported. Let $u\in L^{\infty}(\mathbb{R}\times [0,T))$ be an arbitrary weak solution of (\[scl\]), such that $m(x,t,a)=\partial_t(u\wedge a)+\partial_x f(u\wedge a)$ is locally a Radon measure in ${\R}\times [0,T)\times {\R}$ . Assume the ”entropy production” $m$ satisfies $$\label{minimality}
\int_{\mathbb{R}}\left|m\right|\left(\mathbb{R}\times(0,\bar{t}),a\right)\,da\leq \int_{\mathbb{R}}\left|q\right|\left(\mathbb{R}\times(0,\bar{t}),a\right)\,da~\forall~q\in\mathcal{W}~\text{and}~\forall \bar{t}\in(0,T)\,.$$ Then $u$ is the entropy solution, i.e. satisfies (\[e\_condition\]) and equivalently (\[e\_inequality\]).

A similar criteria in a more restrictive setting is considered by Dafermos in Chapter 9.7 of [@Da]. He considers weak solutions $u$ of (\[scl\]) with initial data $$\label{riemann}
u_0(x)=\left\{\begin{array}{lcl}
                 u_l&\mbox{if}&x<0\,,\\
        u_r&\mbox{if}&x>0\,.
                \end{array}\right.$$ Since the conservation law is invariant under Galilean transformations it is reasonable in this case to consider weak solutions of the form$$u(x,t)=v\left(\frac{x}{t}\right)\,.$$ One can then define $\omega=\frac{x}{t}$ and consider $v$ as a function only dependent of $\omega$, i.e. $v=v(\omega)$. Then $v(\omega)$ satisfies the ordinary differential equation $$\frac{d}{d\omega}\left(f(v(\omega))-\omega v(\omega)\right)+v(\omega)=0$$ in the sense of distributions and has prescribed end states $$\lim_{\omega\rightarrow-\infty}v(\omega)=u_l\quad\mbox{and}\quad\lim_{\omega\rightarrow\infty}v(\omega)=u_r\,.$$ Furthermore it is assumed that $v$ is in $BV$ and 