= Local L1 contraction for scalar conservation laws
{title2=$\int_{a-M(t-s)}^{b+M(t-s)}|u(s)-v(s)|\le\int_{a-Mt}^{b+Mt}|u_0-v_0|$}
If $|f'|\le M$ 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 $W_s+M|W_x|\le0$ 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.
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