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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Regularity of the velocity field for Euler vortex patch evolution
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by Daniel Coutand and Steve Shkoller PDF
Trans. Amer. Math. Soc. 370 (2018), 3689-3720 Request permission


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:
  • Steve Shkoller
  • Affiliation: Department of Mathematics, University of California, Davis, California 95616
  • MR Author ID: 353659
  • Email:
  • 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
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3689-3720
  • MSC (2010): Primary 35Q35, 35Q31
  • DOI:
  • MathSciNet review: 3766863