Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in \R\^3 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 B\^1/3_3,c(\N) . 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 B\^2/3_3,c(\N) conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.
%0 Journal Article
%1 0951-7715-21-6-005
%A Cheskidov, A
%A Constantin, P
%A Friedlander, S
%A Shvydkoy, R
%D 2008
%J Nonlinearity
%K Euler Littlewood-Paley imported
%N 6
%P 1233-1252
%T Energy conservation and Onsager's conjecture for the Euler equations
%U http://stacks.iop.org/0951-7715/21/1233
%V 21
%X Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in \R\^3 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 B\^1/3_3,c(\N) . 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 B\^2/3_3,c(\N) conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.
@article{0951-7715-21-6-005,
abstract = {Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in {\\mathbb R}\^{}3 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 B\^{}{1/3}_{3,c({\\mathbb N})} . 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 B\^{}{2/3}_{3,c({\\mathbb N})} conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.},
added-at = {2008-05-21T09:13:43.000+0200},
author = {Cheskidov, A and Constantin, P and Friedlander, S and Shvydkoy, R},
biburl = {https://www.bibsonomy.org/bibtex/2d5c45a06005b68ce45af5d86647c4ca4/ralfwit},
description = {Energy conservation},
interhash = {53e1884591a5675e9bfc51f51085d108},
intrahash = {d5c45a06005b68ce45af5d86647c4ca4},
journal = {Nonlinearity},
keywords = {Euler Littlewood-Paley imported},
number = 6,
pages = {1233-1252},
timestamp = {2008-05-21T09:13:43.000+0200},
title = {Energy conservation and Onsager's conjecture for the Euler equations},
url = {http://stacks.iop.org/0951-7715/21/1233},
volume = 21,
year = 2008
}