= Gradient catastrophe
A smooth solution undergoes a <gradient catastrophe> when a spatial <derivative> becomes unbounded while the solution itself remains bounded. For $u_t+q'(u)u_x=0$, <characteristics> have $x=s+tq'(u_0(s))$ and $u=u_0(s)$; thus $u_x=u_0'(s)/[1+tq''(u_0(s))u_0'(s)]$. Vanishing of the <characteristic> projection <derivative> gives the first such time. A <weak solution> with an <entropy condition> can continue after this breakdown of the smooth description.
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