Kruzhkov entropy flux
= Kruzhkov entropy flux
{c}
{title2=$q_k(z)=\operatorname{sgn}(z-k)(f(z)-f(k))$}
For $\eta_k(z)=|z-k|$, the displayed flux is continuous at $z=k$ and has <derivative> $\operatorname{sgn}(z-k)f'(z)$ almost everywhere. On a bounded state range with $|f'|\le M$, its symmetric two-state version $Q(a,b)$ obeys $|Q(a,b)|\le M|a-b|$ and is Lipschitz in both arguments. The vanishing factor at $a=b$ removes any sign-function jump contribution.