Blow up and regularity for fractal Burgers equation

By Alexander V. Kiselev, Fedor Nazarov, R. Shterenberg. 2008

199 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0f7e277ee121e3dc641d3e74a61423c2736d9397

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Global well-posedness of slightly supercritical active scalar equations (2012, methodology): “It has been known for a while that in the subcritical case α > 1, global regular solutions exist for sufficiently smooth initial data (see [Resnick 1995] for the SQG equation; the analysis for the Burgers equation is very similar; see, for example, [Kiselev et al. 2008]).”
On a transport equation with nonlocal drift (2014, background): “1), it is known that finite time blowup occurs for any γ < 1 [1, 30, 24, 19].”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, methodology): “For example, this is the case in Burgers equation with fractional diffusion [1, 26], the Keller-Segel model [21], and many other equations [12, 17, 29].”
On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion (2009, background): “For s 1/2 we can show that the equation develops singularities (discontinuities of the derivative) as a corollary of a construction of Kiselev, Nazarov and Shterenberg in [13] for the fractional Burgers equation.”
On the global regularity for the supercritical SQG equation (2014, methodology): “The eventual regularity of weak solutions to supercritical SQG has been previously established in [ 22] for γ sufficiently close to1, using the techniques of [ 1] (see also [ 3]), in [10] for the full rangeγ ∈ (0, 1) by means of the methods devised in [ 17], and finally in [16] for all γ ∈ (0, 1) th”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “There is much work in geostrophic flows, like [188, 174, 175].”