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Blow up and regularity for fractal Burgers equation. (English) Zbl 1186.35020
Authors’ 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.

MSC:
35B44 Blow-up in context of PDEs
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
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