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On the radius of analyticity of solutions to the three-dimensional Euler equations


Authors: Igor Kukavica and Vlad Vicol
Journal: Proc. Amer. Math. Soc. 137 (2009), 669-677
MSC (2000): Primary 76B03, 35L60
DOI: https://doi.org/10.1090/S0002-9939-08-09693-7
Published electronically: September 16, 2008
MathSciNet review: 2448589
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Abstract | References | Similar Articles | Additional Information

Abstract: We address the problem of analyticity of smooth solutions $ u$ of the incompressible Euler equations. If the initial datum is real-analytic, the solution remains real-analytic as long as $ \int_{0}^{t} \left\Vert{\nabla u(\cdot,s)}\right\Vert_{L^\infty} ds< \infty$. Using a Gevrey-class approach we obtain lower bounds on the radius of space analyticity which depend algebraically on $ \exp{\int_{0}^{t} \left\Vert{\nabla u(\cdot,s)}\right\Vert_{L^\infty}}ds$. In particular, we answer in the positive a question posed by Levermore and Oliver.


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Additional Information

Igor Kukavica
Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089
Email: kukavica@usc.edu

Vlad Vicol
Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089
Email: vicol@usc.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09693-7
Received by editor(s): November 13, 2007
Published electronically: September 16, 2008
Additional Notes: Both authors were supported in part by the NSF grant DMS-0604886.
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2008 American Mathematical Society

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