Gradient catastrophe

A smooth solution undergoes a gradient catastrophe when a spatial derivative becomes unbounded while the solution itself remains bounded. For $u_t+q^{\prime }(u)u_x=0$, characteristics have $x=s+t{q}^{\prime }(u_0(s))$ and $u=u_0(s)$; thus $u_x=u_0^{\prime }(s)/[1+t{q}^{\prime \prime }(u_0(s))u_0^{\prime }(s)]$. 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.