\int_{\Gamma_{t_1}^{t_2}}\partial_t\psi\partial_{x}g\,dx\,dt\,.
\end{split}$$ Rewriting the boundary term in (\[int2\]) and using the fact that $\psi(x,t_2)=0$ leads to $$\begin{gathered}
\label{int3}
\int_{\partial\Gamma_{t_1}^{t_2}}g\begin{pmatrix}-\partial_t\psi\\\partial_x\psi\end{pmatrix}\cdot n\,d\sigma=\int_{\Lambda_{t_1}^{t_2}}g_1\begin{pmatrix}-\partial_t\psi\\\partial_x\psi\end{pmatrix}\cdot n\,d\sigma \\=\int_{s_1}^{s_2} g_1(s,\gamma(s,t_1))\left[\partial_t\psi(s,\gamma(s,t_1))\gamma'(s,t_1)+\partial_x\psi(s,\gamma(s,t_{1}))\right]\,ds\,.\end{gathered}$$ Integrating the right-hand side of (\[int3\]) by parts leads to $$\begin{gathered}
\label{int4}
 \int_{s_1}^{s_2} g_1(s,\gamma(s,t_1))\left[\partial_t\psi(s,\gamma(s,t_1))\partial\gamma(s,t_1)+\partial_x\psi(s,\gamma(s,t_{1}))\right]\,ds\\
=\int_{s_1}^{s_2} g_1(s,\gamma(s,t_1))\frac{d}{ds}\psi(s,\gamma(s,t_1))\,ds=-\int_{s_1}^{s_2}\frac{d}{ds}g_1(s,\gamma(s,t_1))\cdot\psi\,ds\,.\end{gathered}$$ Therefore, combining (\[int3\]) and (\[int4\]) we can rewrite the boundary term in (\[int2\]) $$\label{int5}
\int_{\partial\Gamma_{t_1}^{t_2}}g_1\begin{pmatrix}-\partial_t\psi\\\partial_x\phi\end{pmatrix}\cdot n\,d\sigma=\int_{s_1}^{s_2}\left[\partial_x g_1+\partial_{t}g_1\cdot \partial_s\gamma(s,t_1)\right]\psi\,ds$$ Using (\[choice\_of\_t1\]), the right-hand side of (\[int5\]) simplifies to $$\begin{aligned}
\int_{\partial\Gamma_{t_1}^{t_2}}g_1\begin{pmatrix}-\partial_t\psi\\\partial_x\psi\end{pmatrix}\cdot n\,d\sigma&=\int_{s_1}^{s_2}\left[\partial_x g_1-f(\partial_x g_1)\cdot \partial_s\gamma(s,t_1)\right]\psi\,ds\\
&=\int_{\Lambda_{t_1}^{t_2}}\psi\begin{pmatrix}\partial_x g_1\\-f(\partial_x g_1)\end{pmatrix}\cdot \tau\,d\sigma\,,\end{aligned}$$ where $\tau$ is the unit tangent vector of $\Lambda_{t_1}^{t_2}$. We replace now the boundary term in (\[int2\]) using the above identity $$\int_{\Gamma_{t_1}^{t_2}}\partial_x\psi\partial_{t}g\,dx\,dt=\int_{\Lambda_{t_1}^{t_2}}\psi\begin{pmatrix}\partial_x g_1\\-f(\partial_x g_1)\end{pmatrix}\cdot \tau\,d\sigma+ \int_{\Gamma_{t_1}^{t_2}}\partial_t\psi\partial_{x}g\,dx\,dt\,.$$ Finally, this together with (\[int1\]) gives $$\int_{\Lambda_{t_1}^{t_2}}\psi\begin{pmatrix}\partial_x g_1\\-f(\partial_x g_1)\end{pmatrix}\cdot \tau\,d\sigma+ \int_{\Gamma_{t_1}^{t_2}}\partial_t\psi\partial_{x}g+f(\partial_xg)\partial_{x}\psi\,dx\,dt=0\,.$$ Since $v_1=\partial_x g_1$ and by putting $v=\partial_x g$, we see, that $v$ is a solution of (\[modified\_scl\]) in the sense of Definition \[def\_entropy\_solution\]. Finally it remains to show, that $v$ is an entropy solution in the sense, that $$\partial_tv\wedge a+\partial_xf(v\wedge a)\geq0\,.$$ By Corollary 1.7.2 in [@CS] $v$ satisfies for all $(x,t),~(y,t)\in\Gamma_{t_1}^{t_2}$ such that $x<y$ $$v(y,t)-v(x,t)\leq \frac{y-x}{ct}\,.$$ This immediately implies $q(x,t,a)\geq0$ (see Section 8.5 in [@Da]).

Proposition \[existence\] being proved, we now establish some properties for entropy solutions to (\[modified\_scl\]) analogous to those in the classical case (see 