2 Non-linear Conservation Laws and Shocks

For a linear first-order equation the characteristics are fixed in advance, by the coefficients. For a quasilinear equation such as \(u_t + uu_x = 0\) the speed of each characteristic depends on the solution itself, so faster parts of a wave can catch up with slower parts, characteristics cross, and a smooth solution ceases to exist after a finite time: the wave breaks. What happens next cannot be described by a classical solution at all. The resolution, weak solutions with discontinuities called shocks, selected by an entropy condition, is one of the central ideas of the subject, and it governs gas dynamics, traffic flow, river floods and blast waves. This section develops it for scalar equations in one space dimension.

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