  For $0<\alpha<\frac{1}{3}$ we construct unique solutions to the fractal
Burgers equation $\partial_t u + u\partial_xu + (-\Delta)^\alpha u = 0$ which
develop a first shock in finite time, starting from smooth generic initial
data. This first singularity is an asymptotically self-similar, stable $H^6$
perturbation of a stable, self-similar Burgers shock profile. Furthermore, we
are able to compute the spatio-temporal location and H\"older regularity for
the first singularity. There are many results showing that gradient blowup
occurs in finite time for the supercritical range, but the present result is
the first example where singular solutions have been explicitly constructed and
so precisely characterized.

Analysis of PDEs