A Cheskidov et al 2008 Nonlinearity 21 1233 doi:10.1088/0951-7715/21/6/005
Energy conservation and Onsager's conjecture for the Euler equations
A Cheskidov1, P Constantin1, S Friedlander2 and R Shvydkoy2
Show affiliationsRecommended by D Lohse
Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 
conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space 
. We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood–Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to 
conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.
- PACS
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47.10.ab Conservation laws and constitutive relations
- MSC
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76Bxx Incompressible inviscid fluids
- Subjects
- Dates
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Issue 6 (June 2008)
Received 21 November 2007, in final form 1 April 2008
Published 24 April 2008
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Energy conservation and Onsager's conjecture for the Euler equations
A Cheskidov et al 2008 Nonlinearity 21 1233
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