Shock formation and Rankine–Hugoniot

A bounded discontinuous function can still define a weak scalar conservation law because both the state and its flux are locally integrable. Multiply the conservative equation by a smooth compactly supported test function and integrate by parts in space and time. The derivatives then act on the test function, and the prescribed initial state contributes a boundary term at time zero. A weak solution satisfies this identity for every such test, rather than only at points where a classical derivative happens to exist.

The conservative form is crucial. Replacing the flux derivative by a discontinuous coefficient times a distributional state derivative is not an ordinary chain-rule operation at a jump. The weak flux identity instead provides a well-defined distribution without assigning an arbitrary product at the discontinuity. Its local integrals also allow constant left and right Riemann states, even though those states may not have an integrable total mass over the entire line. Local conservation remains meaningful on every finite control interval.

Write $[a]=a_R-a_L$ throughout this lesson. For constant states separated by x=st, represent the state with a Heaviside jump. Its time derivative contributes minus s times the state jump on the moving line, while the spatial flux derivative contributes the flux jump. Their distributional sum vanishes exactly when speed times state jump equals flux jump. Reversing both jump conventions would give the same equation; mixing conventions between numerator and denominator would reverse the physical speed.

For distinct scalar states, the speed is the secant slope of the flux graph. With Burgers' quadratic flux it is the arithmetic mean of the two states, a result derived from conservation rather than guessed from characteristic drawings. For a vector system, Rankine–Hugoniot is a vector equation: one common speed must balance every conserved component. Componentwise scalar averages generally cannot satisfy that requirement, so nonlinear wave curves and family structure enter the system problem.

For a strictly convex flux, a downward jump can gather characteristics arriving from both sides. In the scalar compressive case, the shock speed lies between the left and right characteristic speeds. An upward jump may also satisfy Rankine–Hugoniot algebraically, but its neighboring characteristics spread away and leave a region whose values are not determined by that jump picture. Both can obey the weak conservation identity, which demonstrates why weak balance alone does not establish uniqueness.

An entropy or vanishing-viscosity selection rule excludes the inappropriate expansion jump for the convex scalar problem and replaces it by a rarefaction fan. One should therefore distinguish three checks: whether a classical solution has broken, whether a proposed discontinuity conserves the quantity, and whether it satisfies the appropriate admissibility principle. Numerical schemes have an analogous distinction: a shared interface flux can conserve mass while still converging to an incorrectly selected discontinuity if no suitable entropy mechanism is present.