weak solutions of (\[scl\]) and viscosity subsolutions of $$\label{hamilton_jacobi}
\left. \begin{array}{rcl}
  \partial_t g+f(\partial_x g)&=&0\,,\\
g(x,0)&=&g_0(x)\,.
 \end{array}\right\}$$ Before we are going to prove our assertion, we briefly repeat the definitions of viscosity sub- and supersolutions. We say that $g$ is a viscosity solution of (\[hamilton\_jacobi\]), if for any point $(x_0,t_0)\in\mathbb{R}\times(0,T)$ and for any $\psi \in C^1(\mathbb{R}^2)$ such that $g-\psi$ attains its maximum in $(x_0,t_0)$ the following inequality holds $$\partial_t \psi(x_0,t_0)+f(\partial_x\psi(x_0,t_0))\leq0\,.$$ Similarly we say, that $g$ is a viscosity supersolution of (\[hamilton\_jacobi\]), if for any point $(x_0,t_0)\in\mathbb{R}\times(0,T)$ and for any for any $\psi \in C^1(\mathbb{R}^2)$ such that $g-\psi$ attains its minimum in $(x_0,t_0)$ the following inequality holds $$\partial_t \psi(x_0,t_0)+f(\partial_x\psi(x_0,t_0))\geq0\,.$$ We say that $g$ is a viscosity solution of (\[hamilton\_jacobi\]), if $g$ is both a sub- and supersolution. Theorem 2 in [@CH] establishes a correspondence between weak solutions of (\[scl\]) and viscosity subsolutions of (\[hamilton\_jacobi\]).

\[correspondence\] Let $u\in L^{\infty}(\mathbb{R}\times[0,t))$ be a weak solution of (\[scl\]). Then there exists a $g\in W^{1,\infty}(\mathbb{R}\times[0,T))$ which satisfies (\[hamilton\_jacobi\]) almost everywhere and is such that $u(x,t)=\partial_x g(x,t)$ and $u_0=\partial_x g(x,0)$ for almost every $x\in\mathbb{R}$.

Let $v_1\in L^{\infty}(\mathbb{R}\times[0,T))$ be a weak solution of (\[scl\]); then, according to Theorem \[correspondence\], there exists $g_1\in W^{1,\infty}(\mathbb{R}\times[0,T))$, which solves (\[hamilton\_jacobi\]) almost everywhere. By 