Kruzhkov entropy inequality (${\partial }_t|u-c|+{\partial }_x{q}_c(u)\le 0$)

For a scalar conservation law law $u_t+f(u{)}_x=0$, the Kruzhkov entropy inequalities requirein distributions for every real constant $c$, together with the initial trace. This means the integral against every nonnegative compactly supported test function is nonpositive after applying the distributional derivatives. For a smooth solution the entropy balance is an equality away from $u=c$. For a viscous approximation $u_t+f(u{)}_x=\epsilon u_{xx}$, smooth convex approximations $\eta$ of $|u-c|$ give ${\partial }_t\eta (u)+{\partial }_{x}q(u)=\epsilon {\partial }_{xx}\eta (u)-\epsilon {\eta }^{\prime \prime }(u)u_x^2\le \epsilon {\partial }_{xx}\eta (u)$, with $q^{\prime }={\eta }^{\prime }{f}^{\prime }$. Letting the entropy smoothing vanish and then taking a bounded, strongly locally convergent zero-viscosity limit proves the inequality. The discrete entropy inequality of a monotone conservative scheme is its numerical analogue.