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Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time$
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Philip Isett

Print publication date: 2017

Print ISBN-13: 9780691174822

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691174822.001.0001

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On Onsager's Conjecture

On Onsager's Conjecture

Chapter:
13 On Onsager's Conjecture
Source:
Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
Author(s):

Philip Isett

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691174822.003.0013

This chapter deals with Onsager's conjecture, which would be implied by a stronger form of Lemma (10.1). It considers what could be proven assuming Conjecture (10.1) by turning to Theorem 13.1, which states that for every δ‎ > 0, there exist nontrivial weak solutions (v, p) to the Euler equations on ℝ x ³. Here the energy will increase or decrease in certain time intervals. In order to determine which Hölder norms stay under control during the iteration, the chapter observes that the bound for the spatial derivative of the corrections V and P also controls their full space-time derivative. The chapter also discusses higher regularity for the energy, written as a sum of energy increments.

Keywords:   weak solution, Onsager's conjecture, Euler equations, energy, Hölder norm, spatial derivative, correction, energy increment

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