Asymptotically self-similar shock formation for 1d fractal Burgers equation

Abstract

For $0<\alpha <\frac{1}{3}$ we construct unique solutions to the fractal Burgers equation ${\partial }_{t}u+u{\partial }_{x}u+(-\Delta {)}^{\alpha }u=0$ 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 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 so precisely characterized.

29 pages, corrected minor typos

References in corpus (1)

• Formation of shocks for 2D isentropic compressible Euler

Cited by in corpus (2)

• Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation
• Refined wave breaking for the generalized Fornberg-Whitham equation