2.2 and Section 2.3.

For $0<t_{1}<t_{2}<T$ and a $\delta>0$ we define the set $$\label{the_set_gamma}
\Gamma^{t_{2}}_{t_{1}}:=\{ (x,t) |~t_{2}> t> \gamma(x,t_{1})\}$$ where $$\label{definition_boundary}
\gamma(x,t):=\left\{\begin{array}{ll} t-\hat{\lambda}(x+\delta))&\text{if}~x\leq -\delta\,, \\ 
t&\text{if}~|x|\leq\delta\,,\\ 
t+\hat{\lambda}(x-\delta))&\text{if}~x\geq\delta\,.\end{array}\right.$$ for a constant $0<\hat{\lambda}\leq1$. Further we set $$\Lambda_{t_{1}}^{t_{2}}:=\{(x,t)|~(x,t)=(x,\gamma(x,t_{1}))~\text{and}~t_{1}\leq t<t_{2} \}\,.$$

\[gamma\]

![The set $\Gamma_{t_1}^{t_2}$](the_set_gamma.eps)

As for mentioned we define now a notion of weak respective entropy solution on the domain $\Gamma_{t_{1}}^{t_{2}}$

\[def\_entropy\_solution\] For a $v_{1}\in L^{\infty}(\Lambda_{t_{1}}^{t_{2}})$ we say that $v\in L^{\infty}(\Gamma_{t_{1}}^{t_{2}})$ is weak solution of $$\label{modified_scl}
 \left.\begin{array}{rclcl}
  \partial_{t}v+\partial_{x}f(x)&=&0&\mbox{in}&\Gamma_{t_{1}}^{t_{2}}\,,\\
v&=&v_{1}&\mbox{on}&\Lambda_{t_{1}}^{t_{2}}\,,
 \end{array}\right\}$$ if for all $\psi\in C_{c}^{\infty}(\mathbb{R}\times [0,t_{2}))$ $$\label{weak_formulation}
 \int_{\Gamma_{t_{1}}^{t_{2}}}v\partial_{t}\psi+f(v)\partial_{x}\psi\,dx\,dt+\int_{\Lambda_{t_{1}}^{t_{2}}}
\psi\begin{pmatrix} v_1 \\
 -f(v_1)\end{pmatrix}\cdot\tau\,d\sigma=0$$ holds, where $\tau$ is the unit tangent vector of $\Lambda_{t_{1}}^{t_{2}}$. Furthermore we say that $v\in L^{\infty}(\Gamma_{t_{1}}^{t_{2}})$ is an entropy solution of (\[modified\_scl\]), if $v$ additionally satisfies $$q(x,t,a):=\partial_{t}v\wedge a+\partial_{x}f(v\wedge a)\in\mathcal{M}_{loc}\quad\mbox{and}\quad q(x,t,a)\geq0\,.$$

A priory it is unclear if, for an arbitrary boundary condition $v_{1}\in L^{\infty}(\Lambda_{t_{1}}^{t_{2}})$, the conservation law (\[modified\_scl\]) possess a weak solution or not. We can however prove the following proposition.

\[existence\] Let $v_1\in L^{\infty}(\mathbb{R}\times[0,T))$ be a weak solution of (\[scl\]). Then for all $0<\hat{\lambda}\leq1$ and for almost every $t_{1}\in(0,T)$ and all $t_2\in(t_1,T)$ the problem $$\left.\begin{array}{rclcl}
  \partial_{t}v+\partial_{x}f(v)&=&0&\mbox{in}&\Gamma_{t_{1}}^{t_{2}}\,,\\
v&=&v_1&\mbox{on}&\Lambda_{t_{1}}^{t_{2}}\,,
 \end{array}\right\}$$ has an entropy solution in the sense of Definition \[def\_entropy\_solution\].

The basic idea for proving Proposition \[existence\] is to use the correspondence between 