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Transactions of the American Mathematical Society

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Regularity of the velocity field for Euler vortex patch evolution


Authors: Daniel Coutand and Steve Shkoller
Journal: Trans. Amer. Math. Soc. 370 (2018), 3689-3720
MSC (2010): Primary 35Q35, 35Q31
DOI: https://doi.org/10.1090/tran/7058
Published electronically: November 14, 2017
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Abstract: We consider the vortex patch problem for both the 2-D and
3-D incompressible Euler equations. In 2-D, we prove that for vortex patches with $ H^{k-0.5}$ Sobolev-class contour regularity, $ k \ge 4$, the velocity field on both sides of the vortex patch boundary has $ H^k$ regularity for all time. In 3-D, we establish existence of solutions to the vortex patch problem on a finite-time interval $ [0,T]$, and we simultaneously establish the $ H^{k-0.5}$ regularity of the two-dimensional vortex patch boundary, as well as the $ H^k$ regularity of the velocity fields on both sides of vortex patch boundary, for $ k \ge 3$.


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Additional Information

Daniel Coutand
Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS United Kingdom
Email: d.coutand@ma.hw.ac.uk

Steve Shkoller
Affiliation: Department of Mathematics, University of California, Davis, California 95616
Email: shkoller@math.ucdavis.edu

DOI: https://doi.org/10.1090/tran/7058
Received by editor(s): April 14, 2016
Received by editor(s) in revised form: August 19, 2016
Published electronically: November 14, 2017
Additional Notes: The first author was supported by the Centre for Analysis and Nonlinear PDEs funded by the UK EPSRC grant EP/E03635X and the Scottish Funding Council
The second author was supported by the National Science Foundation under grants DMS-1001850 and DMS-1301380, and by the Royal Society Wolfson Merit Award
Article copyright: © Copyright 2017 American Mathematical Society

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