Entropy condition (${\partial }_t\eta (u)+{\partial }_{x}Q(u)\le 0$)

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 $Q^{\prime }={\eta }^{\prime }{q}^{\prime }$. The displayed inequality is imposed in the distributional sense. Positive diffusion explains this choice: for $u_t+q(u{)}_x=ϵ[D(u)u_x{]}_x$ with $D>0$, the entropy balance is $\eta (u{)}_t+Q(u{)}_x=ϵ[{\eta }^{\prime }(u)D(u)u_x{]}_x-ϵ{\eta }^{\prime \prime }(u)D(u)u_x^2$. 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.