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

2021/05/31 by Chickering, Kyle R., Moreno-Vasquez, Ryan C., Pandya, Gavin · 1 citation
#35L67 #35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.15128

Abstract

For 0<α<(1)/(3) we construct unique solutions to the fractal Burgers equation ∂_t u + u∂_xu + (-Δ)^αu = 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-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.

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• Shock formation in 1D conservation laws I: Inviscid structure

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