Blow up and regularity for fractal Burgers equation

TL;DR

This paper investigates the fractional Burgers equation, establishing conditions for finite-time blow-up and global regularity, and explores solutions with rough initial data, with implications for related equations.

Contribution

It provides new results on blow-up and regularity thresholds for fractional dissipation in Burgers equations, extending to rough initial data and related models.

Findings

Finite time blow-up for $eta < 1/2$

Global existence and analyticity for $eta =1/2$

Existence of solutions with rough initial data in $L^p$

Abstract

The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian $\alpha < 1/2,$ and global existence as well as analyticity of solution for $\alpha \geq 1/2.$ We also prove the existence of solutions with very rough initial data $u_0 \in L^p,$ $1 < p < \infty.$ Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.