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Local Regularity for the Modified SQG Patch Equation
Author(s) -
Kiselev Alexander,
Yao Yao,
Zlatoš Andrej
Publication year - 2017
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21677
Subject(s) - mathematics , plane (geometry) , singularity , mathematical analysis , kernel (algebra) , surface (topology) , euler equations , geometry , pure mathematics
We study the patch dynamics on the whole plane and on the half‐plane for a family of active scalars called modified surface quasi‐geostrophic (SQG) equations. These involve a parameter α that appears in the power of the kernel in their Biot‐Savart laws and describes the degree of regularity of the equation. The values α =0 and α =½ correspond to the two‐dimensional Euler and SQG equations, respectively. We establish here local‐in‐time regularity for these models, for all α ∊ (0,½) on the whole plane and for all small α > 0 on the half‐plane. We use the latter result in [16], where we show existence of regular initial data on the half‐plane that lead to a finite‐time singularity.© 2016 Wiley Periodicals, Inc.