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

  • 1
    $\begingroup$ As usual, what have you tried, etc. $\endgroup$
    – username
    Jul 21, 2022 at 7:45

1 Answer 1


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.

  • $\begingroup$ Thanks. But what does this tell us about $u(x,t)$ alone? $\endgroup$
    – user139844
    Jul 21, 2022 at 10:47
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    $\begingroup$ @Dal: Is $x \mapsto x$ in $BV_{loc}$? Is $BV_{loc}$ a vector space? $\endgroup$
    – Alex M.
    Jul 21, 2022 at 11:14

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