Local Regularity for the Modified SQG Patch Equation |
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| Authors: | Alexander Kiselev Yao Yao Andrej Zlatoš |
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| Affiliation: | 1. Department of Mathematics, Rice University, Houston, TX, USA;2. School of Mathematics Georgia Institute of Technology, Atlanta, GA, USA;3. Department of Mathematics, University of California, La Jolla, CA, USA |
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| Abstract: | 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. |
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