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Double exponential growth of the vorticity gradient for the two-dimensional Euler equation


Author: Sergey A. Denisov
Journal: Proc. Amer. Math. Soc. 143 (2015), 1199-1210
MSC (2010): Primary 76B99; Secondary 76F99
DOI: https://doi.org/10.1090/S0002-9939-2014-12286-6
Published electronically: October 15, 2014
MathSciNet review: 3293735
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Abstract: For the two-dimensional Euler equation on the torus, we prove that the $ L^\infty $-norm of the vorticity gradient can grow as double exponential over arbitrary long but finite time provided that at time zero it is already sufficiently large. The method is based on the perturbative analysis around the singular stationary solution studied by Bahouri and Chemin in 1994. Our result on the growth of the vorticity gradient is equivalent to the statement that the operator of Euler evolution is not bounded in the linear sense in Lipschitz norm for any time $ t>0$.


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Sergey A. Denisov
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: denissov@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12286-6
Keywords: Two-dimensional Euler equation, growth of the vorticity gradient
Received by editor(s): April 4, 2013
Received by editor(s) in revised form: June 5, 2013
Published electronically: October 15, 2014
Additional Notes: This research was supported by NSF grants DMS-1067413, DMS-0758239 and DMS-1159133
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society