\lambda_1=\left(\max\left\{f(-R-1-\alpha),f(R+1+\alpha)\right\}\right)^{-1}\\[5mm]
\mbox{and}\\[5mm]
\alpha=\max\{\|w_1\|_{\infty},\|w_2\|_{\infty}\}\,.\end{array}$$ For the two entropy solutions $v_{1},\, v_{2}$ with boundary conditions $w_{1}$ and $w_{2}$ we consider the kinetic equations $$\left.\begin{array}{rclcl}
 \partial_{t}\chi_{i}+f'(a)\partial_{x}\chi_{i}&=&\partial_{a}q_{i}&\mbox{in}&\mathcal{D}'(\Gamma_{t_{1}}^{t_{2}}\times\mathbb{R})\\
\chi_{i}&=&\chi(w_{i};a)&\mbox{on}&\Lambda_{t_{1}}^{t_{2}}
\end{array}\right\}$$ where $\chi_i=\chi(v_i(x,t);a)$ for $i=1,\,2\,$. Then, as before, we can regularize our kinetic equations with the kernel defined in (\[kernel\]) $$\begin{array}{rclcl}
\partial_{t}\chi^{\varepsilon}_{i}+f'(a)\partial_{x}\chi^{\varepsilon}_{i}&=&\partial_{a}q^{\varepsilon}_{i}(x,t,a)&\mbox{in}&\Gamma_{t_{1}+C\varepsilon}^{t_{2}-C\varepsilon}\\
\end{array}$$ where $$\chi_{i}^{\varepsilon}(x,t,a)=\int_{\Gamma_{t_{1}}^{t_{2}}}\chi(v_{i}(x,t);a)\varphi_{\varepsilon}(x-y,t-s)\,dx\,dt\quad\mbox{for}\quad i=1,\,2$$ and $C>0$ is again chosen such that for $(x,t)\in \Gamma_{t_{1}+C\varepsilon}^{t_{2}-C\varepsilon}$ $$\varphi_{\varepsilon}(x-y,t-s)=0\quad\mbox{for}\quad (y,s)\in\partial\Gamma_{t_1}^{t_2}\,.$$ Then the function $\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}$ satisfies for $(x,t)\in\Gamma_{t_{1}+C\varepsilon}^{t_{2}-C\varepsilon}$ and almost every $a\in\mathbb{R}$ $$\label{regualrized_equation_contraction}
 \partial_{t}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}+f'(a)\partial_{x}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}=\chi^{\varepsilon}_{1}\partial_{a}q_{1}^{\varepsilon}+\chi_{2}^{\varepsilon}\partial_{a}q_{2}^{\varepsilon}
-\chi_{2}^{\varepsilon}\partial_{a}q_{1}^{\varepsilon}-\chi_{1}^{\varepsilon}\partial_{a}q_{2}^{\varepsilon}\,.$$ We make use again of the following abbreviation: $t_1+C\varepsilon=\bar{t}_1$. Let $\bar{t}\in(\bar{t}_1,+t_2-C\varepsilon)$, then we integrate (\[regualrized\_equation\_contraction\]) in $\Gamma_{\bar{t}_1}^{\bar{t}}\times\mathbb{R}$, which leads to $$\begin{gathered}
\label{result1_for_contraction}
 \int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\int_{\mathbb{R}}\partial_{t}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}+f'(a)\partial_{x}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\,da\,dx\,dt\\
=\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\int_{\mathbb{R}}\chi^{\varepsilon}_{1}\partial_{a}q_{1}^{\varepsilon}+\chi_{2}^{\varepsilon}\partial_{a}q_{2}^{\varepsilon}-\chi_{2}^{\varepsilon}\partial_{a}q_{1}^{\varepsilon}-\chi_{1}^{\varepsilon}\partial_{a}q_{2}^{\varepsilon}\,da\,dx\,dt\,.\end{gathered}$$ We recall, that $\chi(v;a)=\mathbbm{1}_{a\leq v}$ and $$q_{1}(x,t,a)=q_{2}(x,t,a)=0\quad\mbox{for}\quad|a|\geq \max\{\|v_{1}\|_{\infty},\|v_{2}\|_{\infty}\}\,.$$ Therefore we can calculate for $(x,t)\in \Gamma_{\bar{t}_1}^{\bar{t}}$ and $i,\,j\in\{1,\,2\}$ $$\begin{aligned}
\int_{\mathbb{R}}\chi_{i}^{\varepsilon}\partial_{a}q_{j}^{\varepsilon}\,da&=\int_{\mathbb{R}}\int_{\Gamma_{t_{1}}^{t_{2}}}\chi(v_{i}(y,s);a)\varphi_{\varepsilon}(x-y,t-s)q_{j}^{\varepsilon}(x,t,a)\,dy\,ds\,da\\
&=\int_{\Gamma_{t_{1}}^{t_{2}}}q_{j}^{\varepsilon}(x,t,v_{i}(y,s))\varphi_{\varepsilon}(x-y,t-s)\,dy\,ds\,.\end{aligned}$$ This implies, since $\varphi_{\varepsilon}$ and $q_{\varepsilon}$ are non-negative $$\int_{\mathbb{R}}\chi_{2}^{\varepsilon}\partial_{a}q_{1}^{\varepsilon}\,da\geq0\quad\mbox{and}\quad  \int_{\mathbb{R}}\chi_{1}^{\varepsilon}\partial_{a}q_{2}^{\varepsilon}\,da\geq0\,,$$ which applied in (\[result1\_for\_contraction\]) leads to the inequality $$\begin{gathered}
\label{result2_for_contraction}
\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\int_{\mathbb{R}}\partial_{t}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}+f'(a)\partial_{x}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\,da\,dx\,dt\\
\leq\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\int_{\mathbb{R}}\chi^{\varepsilon}_{1}\partial_{a}q_{1}^{\varepsilon}+\chi_{2}^{\varepsilon}\partial_{a}q_{2}^{\varepsilon}\,da\,dx\,dt\,.\end{gathered}$$ For the left hand-side of (\[result2\_for\_contraction\]) we compute $$\label{calculation0}\begin{split}
 \int_{\mathbb{R}}\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}&\partial_{t}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}+f'(a)\partial_{x}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\,dx\,dt\,da\\
&=\int_{\mathbb{R}}\int_{\theta^{-}_{\varepsilon}(\bar{t})}^{\theta_{\varepsilon}^{+}(\bar{t})}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(x,\bar{t})\,dx\,da -\int_{\mathbb{R}}\int_{-\delta}^{\delta}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(x,\bar{t}_{1})\,dx\,da\\
&\quad+\hat{\lambda}\int_{\mathbb{R}}\int_{\bar{t}_{1}}^{\bar{t}}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(\theta^{+}_{\varepsilon}(t),t)-\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(\theta_{\varepsilon}^{-}(t),t)\,dt\,da\\
&\quad+\int_{\mathbb{R}}\int_{\bar{t}_{1}}^{\bar{t}}f'(a)\left[\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(\theta_{\varepsilon}^{-}(t),t)-\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(\theta_{\varepsilon}^{+}(t),t)\right]\,dt\,da\,,
\end{split}$$ where $\theta_{\varepsilon}^{\pm}$ are defined in (\[theta\]). After a change of variable this expression simplifies to $$\begin{gathered}
\label{calculation1}
 \int_{\mathbb{R}}\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\partial_{t}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}+f'(a)\partial_{x}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\,dx\,dt\,da\\
=\int_{\mathbb{R}}\int_{\theta^{-}_{\varepsilon}(\bar{t})}^{\theta_{\varepsilon}^{+}(\bar{t})}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(x,\bar{t})\,dx\,da-\int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\int_{\mathbb{R}}\begin{pmatrix}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\\f'(a)\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\end{pmatrix}\cdot \tau\,da\,d\sigma\,.\end{gathered}$$ Using identity (\[calculation1\]) in (\[result2\_for\_contraction\]) gives $$\begin{gathered}
\label{result3_for_contraction}
\int_{\mathbb{R}}\int_{\theta_{\varepsilon}^{-}(\bar{t})}^{\theta^{+}_{\varepsilon}(\bar{t})}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}(x,\bar{t})\,dx\,da\\
\leq\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\int_{\mathbb{R}}\chi^{\varepsilon}_{1}\partial_{a}q_{1}^{\varepsilon}+\chi_{2}^{\varepsilon}\partial_{a}q_{2}^{\varepsilon}\,da\,dx\,dt+\int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\int_{\mathbb{R}}\begin{pmatrix}\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\\f'(a)\left(\chi_{1}^{\varepsilon}-\chi_{2}^{\varepsilon}\right)^{2}\end{pmatrix}\cdot \tau\,da\,d\sigma\,.\end{gathered}$$ We claim $$\label{asymptotic_of_q}
\lim_{\varepsilon\to0^+}\int_{\Gamma_{t_{1}}^{\bar{t}}}\int_{\mathbb{R}}\chi^{\varepsilon}_{i}\partial_{a}q_{i}^{\varepsilon}\,da\,dx\,dt=0\quad\mbox{for}\quad i\in\{1,\,2\}\,.$$ **Proof of Claim (\[asymptotic\_of\_q\]).** We consider the 