the entropy condition

After a shock forms, a conservation law has too many weak solutions, not too few: several different discontinuous functions all satisfy the equation in the integral sense and all obey the Rankine-Hugoniot jump rule. Only one of them is what nature actually does. The entropy condition is the extra rule that throws out the impostors and keeps the physical solution — it encodes the arrow of time that the bare conservation law forgot.

There are several equivalent ways to state it. The Lax entropy condition for a scalar law says a shock from u_L to u_R is admissible only if characteristics RUN INTO it from both sides: f'(u_L) > s > f'(u_R), where s is the shock speed. Geometrically, information must be absorbed by the shock, never created by it; a jump that characteristics flow OUT of (a 'rarefaction shock') is forbidden — that region should spread into a smooth rarefaction instead. Oleinik's condition for a general convex/nonconvex flux is the sharper chord condition on f between u_L and u_R. The deepest version uses entropy/entropy-flux pairs: along the physical solution a convex 'entropy' can only be dissipated, never created, exactly mirroring the second law of thermodynamics.

Why it matters: without it, the theory has no uniqueness, and a single Riemann problem would have infinitely many 'solutions'. The entropy condition restores uniqueness and matches experiment — entropy-violating shocks are never seen in real gases or real traffic. It is also the discrete analogue that good numerical schemes must respect, or they will happily converge to a wrong, entropy-violating answer.