Finite Time Emergence of a Shock Wave for Scalar Conservation Laws

For a convex conservation law ut + f(u)x = 0, u(x, 0) = u0(x), −∞ < x <∞, t > 0, bounded initial data u0(x), are considered that take on constant values u− to the left of a bounded interval, and u+ to the right, with u− > u+. The solution of the initial value problem is shown to collapse in finite time to a single shock wave joining u− to u+. The proof involves comparison with a solution having piecewise constant initial data for which the time of collapse is be calculated explicitly. This result has a significant application to steady granular flow in a chute, and the result is reformulated to apply to the Lighthill-Whitham-Richards equation of traffic flow.