Asymptotically Self-Similar Shock Formation for 1D Fractal Burgers’ Equation
Kyle R. Chickering, Ryan C. Moreno-Vasquez, Gavin Pandya
Source abstract
Abstract. For the case [Formula: see text] we construct unique solutions to the fractal Burgers’ equation [Formula: see text] which develop a first shock in finite time, starting from smooth generic initial data. This first singularity is an asymptotically self-similar, stable [Formula: see text] perturbation of a stable, self-similar Burgers’ shock profile. Furthermore, we are able to compute the spatio-temporal location and Hölder 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.
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