Global well-posedness of slightly supercritical active scalar equations

By Michael Dabkowski, Alexander V. Kiselev, Luis Silvestre, Vlad Vicol. 2012

61 citations recorded by the source corpus.

The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasigeostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6654d1518cb0aa9e78eaf9f733ffc9442f4c0939

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Global well-posedness for the two-dimensional Muskat problem with slope less than 1 (2017, unknown): “See [10], [7] for a good overview of results using this method.”
On a transport equation with nonlocal drift (2014, background): “In comparison, for the fractal Burgers equation, i.e. u = θ in (1.1), it is known that ﬁnite time blowup occurs for any γ 1 [1,19,24,30].”
Finite time singularity for the modified SQG patch equation (2016, background): “The best available result in this direction is global regularity for the logarithmically supercritical SQG equation by Dabkowski, Kiselev, Silvestre, and Vicol [17].”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, methodology): “Inspired by [19], we also explore a slightly supercritical equation.”
On the global regularity for the supercritical SQG equation (2014, background): “bitrarily large initial datum a challenging problem. While for the critical case γ = 1the question of global regularity of (SQGγ) with arbitrarily large datum has been settled [1,6,7,17,19] (see also [11] for the logarithmically supercritical case), the corresponding result for the supercritical equ”