consider now a domain $\Gamma_{t_1}^{t_2}$ such that $\hat{\lambda}\leq\lambda_0$ and an entropy solution $v\in L^{\infty}(\Gamma_{t_1}^{t_2})$ of (\[modified\_scl\]) exists. From Example 1 in [@Is], we know, that $$\label{bound1}
\|v\|_{\infty}\leq R+1\,.$$ Let $\psi\in C^{\infty}_c(\mathbb{R}\times[0,t_2))$. From Theorem 1.3.4 in [@Da] we get for all $\varepsilon>0$ and sufficiently small ($t_1+\varepsilon<t_2$) $$\int_{\Gamma_{t_1+\varepsilon}^{t_2-\varepsilon}}v\partial_t\psi+f(v)\partial_x\psi\,dx\,dt=-\int_{\partial\Gamma_{t_1+\varepsilon}^{t_2-\varepsilon}}\begin{pmatrix}v\\ -f(v)\end{pmatrix}\cdot\tau\psi\,d\sigma\,.$$ Since $\psi(x,t_2)=0$ this implies $$\label{bound_asymp1}
\int_{\Gamma_{t_1+\varepsilon}^{t_2-\varepsilon}}v\partial_t\psi+f(v)\partial_x\psi\,dx\,dt=-\int_{\Lambda_{t_1+\varepsilon}^{t_2-\varepsilon}}\begin{pmatrix}v\\ -f(v)\end{pmatrix}\cdot\tau\psi\,d\sigma\,.$$ As $\varepsilon\to0^+$ the left-hand side of (\[bound\_asymp1\]) converges to $$\int_{\Gamma_{t_1}^{t_2}}v\partial_t\psi+f(v)\partial_x\psi\,dx\,dt\,.$$ Since $v$ is a weak solution of (\[modified\_scl\]) this later fact implies for the right-hand side of (\[bound\_asymp1\]) $$\label{bound_asymp2}
\lim_{\varepsilon\to0^+}\int_{\Lambda_{t_1+\varepsilon}^{t_2-\varepsilon}}\begin{pmatrix}v\\ -f(v)\end{pmatrix}\cdot\tau\psi\,d\sigma=\int_{\Lambda_{t_1}^{t_2}}\begin{pmatrix}v_1\\ -f(v_1)\end{pmatrix}\cdot\tau\psi\,d\sigma$$ In order to keep the notation simple we introduce $$\bar{\gamma}(s)=\begin{pmatrix}s\\\gamma(s,t_1)\end{pmatrix}\quad\mbox{and}\quad v_{\varepsilon}(x,t)=v(x,t+\varepsilon)\,.$$ From (\[bound\_asymp2\]) we deduce $$\label{bound_asymp3}
\lim_{\varepsilon\to0^+}\int_{s_1+\varepsilon}^{s_2-\varepsilon}\left\{v_{1}(\bar{\gamma}(s))-v_{\varepsilon}(\bar{\gamma}(s))+\hat{\lambda}\left[f(v_{\varepsilon}(\bar{\gamma}(s)))-f(v_1(\bar{\gamma}(s))) \right]\right\}\psi\,ds\,.$$ By (\[def\_lambda0\]) we obtain the existence of some constants $C,\,c>0$ for which the following holds $$c\leq 1\pm f(\alpha)\leq C\quad\mbox{for all}\quad\alpha\in\left(-R-1-\|v_1\|_{\infty},R+1+\|v\|_{\infty}\right)\,.$$ Therefore we get from (\[bound\_asymp3\]), that $$\label{bound_asymp4}
\lim_{\varepsilon\to0^+}v(s,\gamma(s,t_1+\varepsilon))=v_1(s,\gamma(s))\quad\mbox{for a.e.}\quad s\in[s_1,s_2]\,.$$ By dominated convergence, we deduce the claim (\[boundary\_asymptotic\]).

To prove the remaining claims of our lemma, we need to introduce the kinetic formulation of conservation laws, we recommend the introduction to this subject given in [@Pe]. However we need here a slight modified version of this formulation. We define for any $v\in\mathbb{R}$ $$\chi(v;a):=\mathbbm{1}_{a\leq v}\, ,$$ where $\mathbbm{1}_{a\leq v}$ is the characteristic function of the set $\{a\in\mathbb{R}:~a\leq 