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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Helicity conservation for the incompressible Euler equations
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by Luigi De Rosa PDF
Proc. Amer. Math. Soc. 148 (2020), 2969-2979 Request permission

Abstract:

In this work we investigate the helicity regularity for weak solutions of the incompressible Euler equations. To prove regularity and conservation of the helicity we will treat the velocity $u$ and its $\operatorname {curl} u$ as two independent functions and we mainly show that the helicity is a constant of motion assuming $u \in L^{2r}_t(C^\theta _x)$ and $\operatorname {curl} u \in L^{\kappa }_t(W^{\alpha ,1}_x)$, where $r,\kappa$ are conjugate Hölder exponents and $2\theta +\alpha \geq 1$. Using the same techniques we also show that the helicity has a suitable Hölder regularity even in the range where it is not necessarily constant.
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Additional Information
  • Luigi De Rosa
  • Affiliation: EPFL SB, Station 8, CH-1015 Lausanne, Switzerland
  • MR Author ID: 1282108
  • Email: luigi.derosa@epfl.ch
  • Received by editor(s): March 15, 2019
  • Received by editor(s) in revised form: November 1, 2019, and November 18, 2019
  • Published electronically: February 26, 2020
  • Communicated by: Catherine Sulem
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 2969-2979
  • MSC (2010): Primary 35Q31, 35A01, 35D30
  • DOI: https://doi.org/10.1090/proc/14952
  • MathSciNet review: 4099784