Let $u\in L^{\infty}(\mathbb{R}\times[0,T))$ be a weak solution of (\[scl\]), such that $$m(x,t,a)=\partial_{t}(u\wedge a)+\partial_x f(u\wedge a)\in\mathcal{M}_{loc}(\mathbb{R}\times\mathbb{R}_+\times\mathbb{R})\,$$ where $\mathcal{M}$ denotes the space of Radon measures. One can define the absolute value of the entropy production over a set $\Omega\subset\mathbb{R}\times\mathbb{R}_+$ as being $$\label{entropy_production}
 EP=\int_{\mathbb{R}}|m|(\Omega,a)\,da\,.$$ In the case of $u$ being an entropy solution and hence in BV, the measure $m(x,t,a)$ and therefore the entropy production of $u$ simplifies to $$\label{ep_entropy_solution}
 EP=\int_{\Omega}\Delta(u^{+},u^{-})\mathcal{H}^1\res J_u\,,$$ where $J_u$ denotes the rectifiable set of jump points of $u$, $u_+$ and $u_-$ are respectively the left and right approximate limits of $u$ for some orientation of $J_u$ and $$\label{Delta}
\Delta(a,b)=\frac{(a-b)^2\left[\frac{f(a)+f(b)}{2}\right]-(a-b)\int_{a}^{b}f(s)\,ds}{\left[(a-b)^2+(f(a)-f(b))^2\right]^{\frac12}}\,.$$ It is natural to compare the different entropic productions of the weak solutions to (\[scl\]) - $BV$ or not $BV$ ! - and to ask the following questions : [*does there exists a weak solution which minimizes the entropy production and, if so, what properties does a minimizer of (\[entropy\_production\]) have*]{}.

In this work we provide a partial answer to this question. We show a weak solution of (\[scl\]) whose entropy production increases in time less, than any other weak solution’s entropy production, has to be the entropy solution. Precisely

Let $\mathcal{W}$ denote the set of defect measures induced 