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On the Helicity conservation for the incompressible Euler equations


Author: Luigi De Rosa
Journal: Proc. Amer. Math. Soc. 148 (2020), 2969-2979
MSC (2010): Primary 35Q31, 35A01, 35D30
DOI: https://doi.org/10.1090/proc/14952
Published electronically: February 26, 2020
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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 $ \mathop {\mathrm {curl}}\nolimits 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 $ \mathop {\mathrm {curl}}\nolimits 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
Email: luigi.derosa@epfl.ch

DOI: https://doi.org/10.1090/proc/14952
Keywords: Incompressible fuid dynamics, Euler equations, helicity conservation, Onsager's conjecture.
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
Article copyright: © Copyright 2020 American Mathematical Society