Fubini’s Theorem we can choose $t_1$ such that both $\partial_t g_1$ and $\partial_x g_1$ are in $L^\infty(\Lambda_{t_1}^{t_2})$ and such that $$\label{choice_of_t1}
 \int_{\Lambda_{t_1}^{t_2}}\partial_{t}g_1+f(\partial_x g_1)\,d\sigma=0\quad\mbox{and}\quad v_1=\partial_x g_1\quad\mbox{a.e. on}\quad\Lambda_{t_1}^{t_2}\,.$$ For $t_1<t_2<T$ we want to show, that there exists a viscosity solution $g$ of $$\label{viscosity_problem}
 \left.\begin{array}{rclcl}
        \partial_t g+f(\partial_x g)&=&0&\mbox{in}&\Gamma_{t_1}^{t_2}\,,\\
g&=&g_1&\mbox{on}&\Lambda_{t_1}^{t_2}\,.
\end{array}\right\}$$ Then we claim, that $v=\partial_x g$ is an entropy solution of (\[modified\_scl\]), in the sense of Definition \[def\_entropy\_solution\]. The existence of such a viscosity solution $g$ will be guaranteed by the existence result of Ishi (see Theorem 3.1 in [@Is]). In order to be able to apply that theorem we must find a viscosity subsolution $\underline{g}$ and a viscosity supersolution $\overline{g}$ of (\[viscosity\_problem\]), which satisfy pointwise $\underline{g}=\overline{g}=g_1$ on $\Lambda_{t_1}^{t_2}$ and $\underline{g}\leq\overline{g}$ in $\Gamma_{t_1}^{t_2}$. According to Proposition 5.1 on page 77 in [@BC], the fact that $g_1$ satisfies (\[hamilton\_jacobi\]) almost everywhere implies, that $g_1$ is a viscosity subsolution of (\[hamilton\_jacobi\]). Thus we can put $\underline{g}=g_1$ and it remains to find a viscosity supersolution $\overline{g}$ such that $\overline{g}\geq g_1$ and $\overline{g}=g_1$ on $\Lambda_{t_{1}}^{t_2}$. For two positive constants $A$, $B$ we consider the function $$\overline{g}_y(x,t)=g_1(y,\gamma(y,t_1))+A|x-y|+B|t-\gamma(y)|\,.$$ We calculate for $(x,t)\in\Gamma_{t_1}^{t_2}$ $$\partial_t\overline{g}_y(x,t)+f(\partial_x \overline{g}_y(x,t))=B\operatorname{sign}(t-\gamma(y))+f(a \operatorname{sign}(x-y))\,.$$ By (\[infinity\_behaviour\]) this is positive, if we choose $A$ large enough. Thus $$\partial_t\overline{g}_y(x,t)+f(\partial_x \overline{g}_y(x,t))>0\quad\mbox{for}\quad 