= Kruzhkov entropy inequality
{c}
{title2=$\partial_t|u-c|+\partial_xq_c(u)\le0$}
For a scalar <conservation law> law $u_t+f(u)_x=0$, the Kruzhkov entropy inequalities require
$$
\partial_t|u-c|+\partial_x\bigl(\operatorname{sgn}(u-c)[f(u)-f(c)]\bigr)\le0
$$
in 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 $u_t+f(u)_x=\varepsilon u_{xx}$, smooth convex approximations $\eta$ of $|u-c|$ give $\partial_t\eta(u)+\partial_xq(u)=\varepsilon\partial_{xx}\eta(u)-\varepsilon\eta''(u)u_x^2\le\varepsilon\partial_{xx}\eta(u)$, with $q'=\eta'f'$. 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.
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