Scalar conservation laws and characteristics

For a smooth scalar u and a twice continuously differentiable flux f, the equation $u_t+\partial_x f(u)=0$ becomes $u_t+f'(u)u_x=0$. Along a curve moving with speed $f'(u)$, the total derivative of u vanishes. Its state is therefore the initial value attached to the curve. The characteristic is a straight line for an autonomous flux depending only on u, because that state and hence its speed remain constant along the smooth characteristic.

This derivation uses the classical chain rule. After a discontinuity forms, the conservative divergence form remains meaningful distributionally, while the product of a discontinuous speed and a distributional derivative requires additional interpretation. One should not silently treat those expressions as equivalent at shocks. The straight-line construction describes the pre-shock classical solution and supplies information for building a weak continuation, but it does not by itself select that continuation.

Differentiate the characteristic position with respect to its initial label. The resulting Jacobian is one plus time times the flux curvature times the initial slope. While it is positive and the map is globally invertible, the spatial slope equals the initial slope divided by this Jacobian. For a strictly convex flux, a negative initial slope makes characteristics converge, while a positive slope spreads them. The sign concerns the product of flux curvature and profile slope, not merely whether the state is positive or negative.

For Burgers' flux, the speed is u and the curvature is one. If bounded continuously differentiable initial data have bounded derivative with an attained negative minimum, the first positive Jacobian zero occurs at minus the reciprocal of that minimum. This gives the first characteristic gradient-breaking time in this whole-line construction. The states remain in their original bounded range before breaking because they are constant along characteristics. What blows up is their spatial derivative as neighboring labels compress into the same position.

When neighboring speeds separate, the solution can remain smooth or form a continuous fan from discontinuous increasing data. When they collide, continuing every labeled line produces multiple values at one position. A conserved density cannot take all those values simultaneously. Replacing the overlap by a moving jump restores a single-valued weak description, but its speed must be determined from conservation rather than from an arbitrary average chosen for visual smoothness.

The characteristic picture also identifies a finite local propagation bound when the solution stays in a bounded state interval and the flux derivative is bounded there. This scalar whole-line statement is different from a universal speed independent of data: changing the allowed state range may change the maximal speed. Source terms, explicit spatial flux dependence, boundaries, and coupled systems modify the characteristic equations. Each extension needs its own hypotheses before the simple initial-label formulas can be reused.