= Entropy flux for a scalar conservation law
{title2=$q'=\eta'f'$}
A convex entropy $\eta$ is paired with any flux antiderivative satisfying $q'(z)=\eta'(z)f'(z)$ almost everywhere. Its admissibility inequality is $\partial_t\eta(u)+\partial_xq(u)\le0$ distributionally, with the matching initial entropy trace. Affine entropies $\eta(z)=\pm z$ recover the <weak formulation>, while bounded classical solutions satisfy equality for smooth entropies and then by approximation for piecewise smooth convex entropies.
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