v\}$. Then a weak solution $v\in L^{\infty}(\Gamma_{t_{1}}^{t_{2}})$ of (\[modified\_scl\]) satisfies in the distributional sense $$\label{kinetic_formulation}
\left.\begin{array}{rclcl}
\partial_{t}\chi(v(x,t);a)+f'(a)\partial_{x}\chi(v(x,t);a)&=&\partial_{a}q(x,t,a)&\mbox{in}&\Gamma_{t_1}^{t_2}\,,\\
\chi(v;a)&=&\chi(v_1;a)&\mbox{on}&\Lambda_{t_1}^{t_2}\,.
\end{array}\right\}$$ In other words this means, that for all $\psi\in C_c^{\infty}(\mathbb{R}\times[0,t_2)\times\mathbb{R})$ $$\begin{gathered}
 \int_{\Gamma_{t_1}^{t_2}}\int_{\mathbb{R}}\chi(v;a)\partial_t\psi+f'(a)\chi(v;a)\partial_x\psi\,da\,dx\,dt\\=\int_{\Gamma_{t_1}^{t_2}}+\int_{\Lambda_{t_1}^{t_2}}\int_{\mathbb{R}}\psi\begin{pmatrix}\chi(v1;a)\\-f'(a)\chi(v_1;a)\end{pmatrix}\cdot\tau\,da\,d\sigma\,.\end{gathered}$$ In order to prove (\[contraction\_inequality\]), (\[Linfty\_bound\]) and (\[control\_of\_the\_defect\_measure\]) we need to regularize our kinetic equation (\[kinetic\_formulation\]). We choose $\varphi_{1}(x),~\varphi_{2}(t)\in C^{\infty}_{c}(\mathbb{R})$ non-negative functions such that $$\mbox{supp}~\varphi_1\subset(-1,1)\,,\quad\mbox{supp}~\varphi_{2}\subset[-1,0]$$and$$\int_{\mathbb{R}}\varphi_{2}\,dx=\int_{\mathbb{R}}\varphi_{1}\,dx=1\,.$$ We define the kernel $$\label{kernel}
\varphi_{\varepsilon}(x,t)=\frac{1}{\varepsilon^2}\varphi_1\left(\frac{x}{\varepsilon}\right)\varphi_2\left(\frac{t}{\varepsilon}\right)\,.$$ For a constant $C$ depending only from $\hat{\lambda}$ we have $$\operatorname{dist}((x,t),\partial\Gamma_{t_1}^{t_2})>\varepsilon\quad\mbox{for all}\quad(x,t)\in\Gamma_{t_1+C\varepsilon}^{t_2-C\varepsilon}\,.$$ Consequently for $(x,t)\in\Gamma_{t_{1}+C\varepsilon}^{t_{2}-C\varepsilon}$ $$\label{zero_on_boundary}
 \varphi_{\varepsilon}(x-y,t-s)=0\quad\mbox{for}\quad(y,s)\in\partial\Gamma_{t_{1}}^{t_{2}}\,.$$ We define moreover the two mollified functions $$\chi_{\varepsilon}(x,t,a)=\int_{\Gamma_{t_{1}}^{t_{2}}}\varphi_{\varepsilon}(x-y,t-s)\chi(v(y,s);a)\,dy\,ds$$ and $$q_{\varepsilon}(x,t,a)=\int_{\Gamma_{t_{1}}^{t_{2}}}\varphi_{\varepsilon}(x-y,t-s)q(y,s,a)\,dy\,ds\,.$$ For $q_{\varepsilon}$ and $(x,t)\in\Gamma_{t_1+C\varepsilon}^{t_2-C\varepsilon}$ we compute $$\begin{aligned}
q_{\varepsilon}(x,t,a)- q_{\varepsilon}(x,t,b)&=\int_{\Gamma_{t_{1}}^{t_{2}}}\varphi_{\varepsilon}(x-y,t-s)\left[q(y,s,a)-q(y,s,b)\right]\,dy\,ds\\
&=\int_{\Gamma_{t_{1}}^{t_{2}}}\partial_{t}\varphi_{\varepsilon}(x-y,t-s)\left[v\wedge a-v\wedge b\right]\\
&\quad+\partial_{x}\varphi_{\varepsilon}(x-y,t-s)\left[f(v\wedge a)-f(v\wedge b)\right]\,dy\,ds\,,\\\end{aligned}$$ where we have made use of (\[zero\_on\_boundary\]). Since $$|v\wedge a-v\wedge b|\leq\|v\|_{\infty}|b-a|$$ it follows from the calculation above $$\begin{aligned}
 \left|q_{\varepsilon}(x,t,a)- q_{\varepsilon}(x,t,b)\right|&\leq \int_{\Gamma_{t_{1}}^{t_{2}}}\left|\partial_{t}\varphi_{\varepsilon}\right|\cdot\|v\|_{\infty}|b-a|+C\left|\partial_{x}\varphi_\varepsilon\right|\cdot\|v\|_{\infty}|b-a|\,dy\,ds\\
&\leq C|b-a|\,.\end{aligned}$$ Therefore $q_{\varepsilon}$ is Lipschitz continuous with respect to the kinetic variable $a$ and we have for almost every $a\in\mathbb{R}$ in the classical sense $$\label{regularized_equation}
\partial_{t}\chi_{\varepsilon}+f'(a)\partial_{x}\chi_{\varepsilon}=\partial_{a}q_{\varepsilon}(x,t,a)\quad\mbox{in}\quad\Gamma_{t_1+C\varepsilon}^{t_2-C\varepsilon}\,.$$ Notice that due to the convolution with $\varphi_{\varepsilon}$ both $\chi_{\varepsilon}$ and $q_{\varepsilon}$ are smooth with respect to $(x,t)$. Furthermore for $(x,t)\in\Gamma_{t_1+C\varepsilon}^{t_2-C\varepsilon}$ the function $q_{\varepsilon}$ satisfies $$\label{compact_support_of_q}
 q_{\varepsilon}(x,t,a)=0\quad\mbox{if}\quad |a|\geq \|v\|_{\infty}\,.$$ This follows from the classical fact, that $$q(x,t,a)=0\quad\mbox{for}\quad |a|\geq \|v\|_{\infty}\,.$$ Indeed for $|a|\geq \|v\|_{\infty}$ and $\psi\in C^{\infty}_c(\Gamma_{t_1}^{t_2})$ we compute 