ASYMPTOTICALLY SELF-SIMILAR SHOCK FORMATION FOR 1D FRACTAL BURGERS' EQUATION\ast

\partial tu + 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 H6 perturbation of a stable, self-similar Burgers' shock profile. Furthermore, we are able to compute the spatio-temp oral location and Ho"\lder 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 thus precisely characterized.