 Kato inequality for scalar conservation laws

For bounded entropy solutions of the same scalar law, $Q(u,v)=\mathrm{sgn}(u-v)(f(u)-f(v))$ satisfies the displayed distributional inequality, with initial value $|u_0-v_0|$ in its test-function form. The doubling of variables for scalar conservation laws, local translation continuity and the averaged initial trace of an entropy solution prove it. Its significance is comparison between two solutions rather than an inequality against a constant state.