  This paper studies the smoothing effect for entropy solutions of conservation
laws with general nonlinear convex fluxes on $\mathbb{R}$. Beside convexity, no
additional regularity is assumed on the flux. Thus, we generalize the
well-known $\mathrm{BV}$ smoothing effect for $\mathrm{C}^2$ uniformly convex
fluxes discovered independently by P. D. Lax and O. Oleinik, while in the
present paper the flux is only locally Lipschitz. Therefore, the wave velocity
can be dicontinuous and the one-sided Oleinik inequality is lost. This
inequality is usually the fundamental tool to get a sharp regularizing effect
for the entropy solution. We modify the wave velocity in order to get an
Oleinik inequality useful for the wave front tracking algorithm. Then, we prove
that the unique entropy solution belongs to a generalized $\mathrm{BV}$ space,
$\mathrm{BV}^\Phi$.
