= Averaged initial trace of an entropy solution
{title2=$\delta^{-1}\int_0^\delta\|u(t)-u_0\|_{L^1(K)}dt\longrightarrow0$}
For a bounded <entropy solution> with its initial entropy inequalities, the displayed limit holds on every compact spatial set. Test a constant-level entropy with $(1-t/\delta)_+\rho(x)$. The flux remainder is $O(\delta)$, giving a bound by $\int\rho|u_0-k|$. A finite smooth partition and constant levels approximate the initial function in local $L^1$; the triangle inequality makes the limsup arbitrarily small. This averaged trace suffices for one-sided time mollifiers supported in $[\delta,2\delta]$.
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