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Regularizing effect of the forward energy cascade in the inviscid dyadic model


Authors: Alexey Cheskidov and Karen Zaya
Journal: Proc. Amer. Math. Soc. 144 (2016), 73-85
MSC (2010): Primary 35Q31, 76B03
DOI: https://doi.org/10.1090/proc/12494
Published electronically: September 11, 2015
MathSciNet review: 3415578
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Abstract: We study the inviscid dyadic model of the Euler equations and prove some regularizing properties of the nonlinear term that occur due to forward energy cascade. We show every solution must have $ \frac {3}{5}$ $ L^2$-based (or $ \frac {1}{10}$ $ L^3$-based) regularity for all positive time. We conjecture this holds up to Onsager's scaling, where the $ L^2$-based exponent is $ \frac {5}{6}$ and the $ L^3$-based exponent is $ \frac {1}{3}$.


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Alexey Cheskidov
Affiliation: Department of Mathematics, Statistics, and Mathematical Computer Science, University of Illinois at Chicago, 851 South Morgan Street (M/C 249), Chicago, Illinois 60607
Email: acheskid@uic.edu

Karen Zaya
Affiliation: Department of Mathematics, Statistics, and Mathematical Computer Science, University of Illinois at Chicago, 851 South Morgan Street (M/C 249), Chicago, Illinois 60607
Email: kzaya2@uic.edu

DOI: https://doi.org/10.1090/proc/12494
Keywords: Dyadic model, regularity, Euler equations, Onsager's conjecture
Received by editor(s): October 28, 2013
Received by editor(s) in revised form: November 13, 2013, February 3, 2014, February 6, 2014, and February 28, 2014
Published electronically: September 11, 2015
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

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