 Local L1 contraction for scalar conservation laws

If $|f^{\prime }|\le M$ on the solutions' state range, the Kato inequality for scalar conservation laws implies the displayed estimate for almost every $s\in [0,t]$. Approximate the shrinking interval by smooth cutoffs $W$ satisfying $W_s+M|W_x|\le 0$ and multiply by a temporal cutoff ending at $s$. Since $|Q|\le M|u-v|$, the lateral flux cannot increase the integral. The estimate gives uniqueness and a finite propagation speed without global $L^1$ assumptions on the data.