On Nonperiodic Euler Flows with Hölder Regularity
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205_2016_Article_973.pdf
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Author(s)
Oh, Sung-Jin
Isett, Philip
Date Issued
February 2016
Journal
Archive for Rational Mechanics and Analysis
Publisher
Springer Berlin Heidelberg
Citation
Isett, Philip, and Sung-Jin Oh. “On Nonperiodic Euler Flows with Hölder Regularity.” Archive for Rational Mechanics and Analysis 221, no. 2 (February 24, 2016): 725–804.
Version
Author's final manuscript
Abstract
In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/31/3 . This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L[∞ over t]B[1/3 over 3,∞] due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/51/5 . The main result of the present paper shows that any given smooth Euler flow can be perturbed in C[1/5−ϵ over t,x] on any pre-compact subset of R×R[superscript 3] to violate energy conservation. Furthermore, the perturbed solution is no smoother than C[1/5−ϵ over t,x]. As a corollary of this theorem, we show the existence of nonzero C[1/5−ϵ over t,x] solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00205-016-0973-3
