Blow up and regularity for fractal Burgers equation

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 < 1/2, and global existence as well as analyticity of solution for 1/2. We also prove the existence of solutions with very rough initial data u 0 L p , 1 < p < . Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation. Contents 1. Introduction 211 2. Local existence, uniqueness and regularity 214 3. Global existence and analyticity for the critical case = 1/2 220 4. Blow-up for the supercritical case. 225 5. Global existence and regularity for rough initial data for the case = 1/2 233 6. The critical Sobolev space 237