Consider the scalar conservation law $$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$ where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>c>0$).

The solution can be shown to satisfy so-called "Oleinik's entropy condition": $$ \frac{u(x+a,t)-u(x,t)}{a} \leq \frac{c}{t} \hspace{0.7 cm} a>0,t>0.$$

Question: How does this condition imply $u(\cdot,t) \in BV_{loc}(\mathbb R)$?

Consider the scalar conservation law $$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$ where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>c>0$).

The solution can be shown to satisfy so-called "Oleinik's entropy condition": $$ \frac{u(x+a,t)-u(x,t)}{a} \leq \frac{c}{t} \hspace{0.7 cm} a>0,t>0.$$

Question: How does this condition imply $u(\cdot,t) \in BV_{\mathrm{loc}}(\mathbb R)$?