Conservation laws and shocks

A scalar conservation law has form u_t+∂_x f(u)=0. Integrating over a fixed interval says its total amount changes by incoming flux minus outgoing flux. Smooth solutions obey u_t+f'(u)u_x=0, so characteristics move at speed f'(u). The conservative form remains meaningful when u is only bounded and derivatives are distributions. One tests against smooth compactly supported functions in space-time and integrates by parts, retaining an initial-data term when tests reach the initial line. Replacing this with an informal product u u_x at a jump would lose that definition.

Suppose a discontinuity separates constant left and right states and moves along x=s t. Conservation across a shrinking space-time control volume gives s(u_R−u_L)=f(u_R)−f(u_L). This Rankine–Hugoniot relation fixes its speed whenever the states differ. For Burgers flux f(u)=u²/2, the speed is their arithmetic mean. It is the speed of the discontinuity, not the characteristic speed of either constant state. Across more general moving curves the same relation holds locally with the instantaneous interface speed.

For a smooth solution, any differentiable entropy η(u) has a companion flux q with q'=η'f', and η_t+q_x=0 follows from the chain rule. For admissible weak solutions, convex entropies instead satisfy an inequality η_t+q_x≤0 in the distributional sense. For a bounded scalar solution with a sufficiently smooth flux, requiring the appropriate full family of entropy inequalities and initial-data attainment supplies a uniqueness framework. One chosen quadratic entropy is a useful calculation but should not be advertised as a universal substitute for that full theorem in every flux or solution class.

For strictly convex Burgers flux, a jump with u_L>u_R has characteristics entering it from both sides, with u_L>s>u_R. It is a compressive entropy shock. If u_L<u_R, a jump can still satisfy Rankine–Hugoniot but its characteristics separate; the entropy solution is instead a rarefaction fan connecting the states. Adding a small positive viscosity and taking its controlled vanishing limit is another way to motivate this selection. Numerical viscosity must also be used carefully: excessive diffusion can smear the front, and a method must remain consistent with the intended conservation law.

The conserved quantity remains governed by the conservative flux even when a convex mathematical entropy decreases. These statements are compatible because they measure different functions of u. With periodic or otherwise closed conditions, the integral of u remains fixed, while the integral of u²/2 can decrease at a Burgers shock. For gas dynamics the word entropy is tied to thermodynamic conventions, which can use a sign opposite to a convex mathematical entropy. It is safer to display the exact inequality and flux pair than to rely on the word alone.

The spectral lab can illustrate the challenge of representing fine structure: increase N at a low decay p and notice the sharper variations from high-frequency modes; raise p to suppress them. The fixed family Σ sin(nx−t)/nᵖ is smooth for every finite N and has no Burgers update or entropy condition. It does not form a shock, calculate a Riemann solution, or show numerical convergence to an entropy solution. The piecewise formulas in the example and practice supply the actual shock and rarefaction models, whose speeds can be checked directly.