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 $u(\cdot,t) \in BV\_{\mathrm{loc}}(\mathbb R)$?



Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law
The function $f(x):=u(x,t)-cx/t$ decreases, thus $u(x,t)=f(x)+cx/t$ is a sum of two monotone functions, so belongs to local BV.


