= Entropy condition
{title2=$\partial_t\eta(u)+\partial_xQ(u)\leq0$}
For a scalar <conservation law> $u_t+q(u)_x=0$, an entropy condition restricts its <weak solutions> to physically admissible ones. For every convex entropy function $\eta$, define an entropy flux by $Q'=\eta'q'$. The displayed inequality is imposed in the distributional sense. Positive <diffusion> explains this choice: for $u_t+q(u)_x=\epsilon[D(u)u_x]_x$ with $D>0$, the entropy balance is $\eta(u)_t+Q(u)_x=\epsilon[\eta'(u)D(u)u_x]_x-\epsilon\eta''(u)D(u)u_x^2$. The last term is nonpositive, and an appropriate vanishing-diffusion limit yields the entropy inequality. For a strictly concave flux and a single <shock>, this selects states with incoming <characteristics>, rather than a discontinuity whose characteristics spread outward.
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