Energy conservation and Onsager's conjecture for the Euler equations TeX Export

@misc{citeulike:1215608, abstract = {Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of 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(\NN)}\$. 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(\NN)}\$ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.}, archivePrefix = {arXiv}, author = {Cheskidov, A. and Constantin, P. and Friedlander, S. and Shvydkoy, R.}, citeulike-article-id = {1215608}, citeulike-linkout-0 = {http://arxiv.org/abs/0704.0759}, citeulike-linkout-1 = {http://arxiv.org/pdf/0704.0759}, day = {5}, eprint = {0704.0759}, keywords = {3d, euler-equation, onsager, regularity}, month = {Apr}, posted-at = {2007-04-08 11:31:55}, priority = {2}, title = {Energy conservation and Onsager's conjecture for the Euler equations}, url = {http://arxiv.org/abs/0704.0759}, year = {2007} }

TY - ELEC ID - citeulike:1215608 L3 - citeulike-article-id:1215608 N2 - Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of 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 $B^{1/3}_{3,c(\NN)}$. 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 $B^{2/3}_{3,c(\NN)}$ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range. TI - Energy conservation and Onsager's conjecture for the Euler equations KW - 3d KW - euler-equation KW - onsager KW - regularity AU - Cheskidov, A AU - Constantin, P AU - Friedlander, S AU - Shvydkoy, R PY - 2007/Apr/05/ UR - http://arxiv.org/abs/0704.0759 ER -