denotes $J_v$ the set of jump points $\omega$ for $v$. For a given entropy-entropy flux pair $(\eta(u),\xi(u))$ C. Dafermos defines the combined entropy of the shocks in $v$ by $$\label{combined_entropy}
\mathcal{P}_{v}=\sum_{\omega\in J_v}\left\{\xi(v(\omega+))-\xi(v(\omega-))-\omega\left[\eta(v(\omega+))-\eta(v(\omega-))\right]\right\}\,.$$ Furthermore he introduces the rate of change of the total entropy production $$\dot{\mathcal{H}}_v=\frac{d}{dt}\int_{-\infty}^{\infty}\eta(u(x,t))\,dx=\int_{-\infty}^{\infty}\eta(v(\omega))\,d\omega\,,$$ for entropy-entropy flux pairs $(\eta,\xi)$ such that $\eta(u_l)=\eta(u_r)=0$.

He shows that in this simple case the rate of change of the total entropy and the entropy productions are related to each other by $$\dot{\mathcal{H}}_v=\mathcal{P}_v+\xi(u_l)-\xi(u_r)\,.$$ We can now relate the combined entropy $\mathcal{P}_v$ to our entropy productions (\[entropy\_production\]). To do so one notices, that for a $T>0$, $\psi\in C^{\infty}_{c}(\mathbb{R}\times(0,T))$ and an entropy-entropy flux pair $(\eta,\xi)$ we get after a change of variable $$\label{relation_entropy_production}
\begin{split}
\int_{\mathbb{R}\times[0,T]}&\eta(u(x,t))\partial_t\psi+\xi(u(x,t))\partial_x\psi\,dx\,dt\\
&=\int_{\mathbb{R}}\xi(v(\omega))\frac{d}{d\omega}\phi(\omega)-\eta(\omega)\frac{d}{d\omega}\left[\omega\cdot\phi(\omega)\right]\,d\omega\\
&=\sum_{\omega\in J_v}\phi(\omega)\left\{\xi(v(\omega+))-\xi(v(\omega-))-\omega\left[\eta(v(\omega+))-\eta(v(\omega-))\right]\right\}\,,
\end{split}$$ where $$\label{def_phi}
\phi(\omega)=\int_0^T\psi(\omega t,t)\,dt\,.$$ For a jump point $\omega$ we write $v_+=v(\omega+)$ and $v_-=v(\omega-)$, then taking the particular entropy-entropy flux pair $(\eta_a,\xi_a)$, defined in (\[kruzhkov\_family\]) and using identity (\[relation\_entropy\_production\]) gives $$\begin{gathered}
\label{relation_entropy_production1}
\int_{\mathbb{R}}\int_{\mathbb{R}\times[0,T]}\psi(x,t)dm(x,t,a)\,da\\
=-\int_{\mathbb{R}}\sum_{\omega\in J_v}\phi(\omega)\left\{\xi_a(v_+)-\xi_a(v_-))-\omega\left[\eta_a(v_+)-\eta_a(v_-)\right]\right\}\,da\\\end{gathered}$$ A short calculation reveals $$\begin{gathered}
\label{relation_entropy_production2}
\int_{\mathbb{R}}\xi_a(v_+)-\xi_a(v_-)-\omega\left[\eta_a(v_+)-\eta_a(v_-)\right]\,da\\=-\frac12\omega\left(v_-^2-v_+^2\right)-\int_{v_-}^{v_+}sf'(s)\,ds\,,\end{gathered}$$ where we used the Rankine-Hugoniot condition for self similar solutions: $$f(v_+)-f(v_-)=\omega(v_+-v_-)\,.$$ Applying (\[relation\_entropy\_production2\]) in (\[relation\_entropy\_production1\]) gives $$\begin{gathered}
\label{relation_entropy_production3}
\int_{\mathbb{R}}\int_{\mathbb{R}\times[0,T]}\psi(x,t)dm(x,t,a)\,da\\
=\sum_{\omega\in J_v}\phi(\omega)\left\{\xi(v(\omega+))-\xi(v(\omega-))-\omega\left[\eta(v(\omega+))-\eta(v(\omega-))\right]\right\}\,da\,,\end{gathered}$$ where $(\eta,\xi)$ is the entropy-entropy flux pair $$\label{part_entropy_pair}
\eta(v)=\frac12v^2\quad\mbox{and}\quad \xi(v)=\int_0^v sf'(s)\,ds\,.$$ From (\[relation\_entropy\_production3\]) we 