an entropy-flux pair

The entropy condition can be stated as a geometric rule about characteristics, but there is a more flexible and far-reaching way to say it: in terms of an extra conserved-looking quantity that, on the true solution, can only decrease. An entropy-flux pair is precisely such a bookkeeping pair — a quantity eta and its companion flux q designed so that smooth flow conserves eta but a shock destroys it.

For a scalar law u_t + f(u)_x = 0, a pair (eta, q) of functions of u is an entropy-flux pair if q'(u) = eta'(u) f'(u). The point of this compatibility relation: whenever u is a smooth solution, the chain rule gives eta(u)_t + q(u)_x = 0 exactly — eta is an extra conserved quantity for free. Now take eta to be CONVEX. Then for the physically correct (entropy) solution, even across shocks, one has the inequality eta(u)_t + q(u)_x <= 0 in the distributional sense: the entropy is not conserved but DISSIPATED at shocks. This single inequality, required to hold for every convex entropy pair, is the cleanest statement of admissibility. Kruzhkov sharpened it further by using the special family eta(u) = |u - k| with q(u) = sign(u - k)(f(u) - f(k)) for every constant k, which is enough to pin down a unique solution.

Why it matters: entropy pairs turn the entropy condition into a robust inequality that survives passing to limits and works in any number of space dimensions, which is exactly what is needed to prove existence and uniqueness. They also give numerical analysts a quantity to monitor: an entropy-stable scheme is one that dissipates a discrete entropy, guaranteeing it cannot converge to a wrong weak solution. For systems, however, a convex entropy need not exist — when it does (as for gas dynamics, where physical entropy works), it is a powerful structural gift.