= Doubling of variables for scalar conservation laws
Apply the <entropy solution> inequality for $u(t,x)$ with level $v(s,y)$ and for $v(s,y)$ with level $u(t,x)$, integrate both in the other pair of variables, and add them. Symmetry of the <Kruzhkov entropy flux> produces <derivatives> $\partial_t+\partial_s$ and $\partial_x+\partial_y$. A nonnegative mollifier concentrated near the diagonal then yields the <Kato inequality for scalar conservation laws>. Boundary-time terms require an initial trace argument, not merely interior translation continuity.
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