Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 31B / a

Characteristics beginning at $\xi$ have $x=\xi +t{u}_I(\xi )$ and carry the constant value $u_I(\xi )$. In the sloping interval, $x=t+(1-t)\xi$. For $0\le t<1$ the resulting single-valued solution isIt is continuous and piecewise smooth, locally Lipschitz for $t<1$, but not a classical $C^1$ solution across the two derivative corners. Its weak derivatives satisfy the equation, since the function itself has no jumps there. The sloping gradient is $-1/(1-t)$ and blows up at $t=1$, when those characteristics coalesce.

Thus the continuous characteristic description has maximal interval $0\le t<1$. If “weak solution” is taken literally, it does not cease to exist then: the entropy continuation is a shock with left/right states 1 and 0 and Rankine-Hugoniot speed $1/2$, situated at $x=(t+1)/2$ for $t\ge 1$. It is a global entropy weak solution, with the step at $t=1$ as its limiting trace. This distinguishes breakdown of the characteristic parametrization from nonexistence of a weak continuation.