= Kato inequality for scalar conservation laws
{c}
{title2=$\partial_t|u-v|+\partial_xQ(u,v)\le0$}
For bounded <entropy solutions> of the same scalar law, $Q(u,v)=\operatorname{sgn}(u-v)(f(u)-f(v))$ 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.
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