Blow-up in finite time for the dyadic model of the Navier-Stokes equations TeX

@misc{citeulike:456147, abstract = {We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlovi\'c. They showed a finite time blow-up in the case where the dissipation degree \$\alpha\$ is less than 1/4. In this paper we prove the existence of weak solutions for all \$\alpha\$, energy inequality for every weak solution with nonnegative initial datum starting from any time, local regularity for \$\alpha \> 1/3\$, and global regularity for \$\alpha \geq 1/2\$. In addition, we prove a finite time blow-up in the case where \$\alpha\<1/3\$. It is remarkable that the model with \$\alpha=1/3\$ enjoys the same estimates on the nonlinear term as the 4D Navier-Stokes equations. Finally, we discuss a weak global attractor, which coincides with a maximal bounded invariant set for all \$\alpha\$ and becomes a strong global attractor for \$\alpha \geq 1/2\$.}, archivePrefix = {arXiv}, author = {Cheskidov, Alexey}, citeulike-article-id = {456147}, eprint = {math/0601074}, keywords = {blow-up, navier-stokes}, month = {Jan}, posted-at = {2006-01-05 11:05:44}, priority = {2}, title = {Blow-up in finite time for the dyadic model of the Navier-Stokes equations}, url = {http://arxiv.org/abs/math/0601074}, year = {2006} }

TY - ELEC ID - citeulike:456147 L3 - citeulike-article-id:456147 N2 - We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlovi\'c. They showed a finite time blow-up in the case where the dissipation degree $\alpha$ is less than 1/4. In this paper we prove the existence of weak solutions for all $\alpha$, energy inequality for every weak solution with nonnegative initial datum starting from any time, local regularity for $\alpha > 1/3$, and global regularity for $\alpha \geq 1/2$. In addition, we prove a finite time blow-up in the case where $\alpha<1/3$. It is remarkable that the model with $\alpha=1/3$ enjoys the same estimates on the nonlinear term as the 4D Navier-Stokes equations. Finally, we discuss a weak global attractor, which coincides with a maximal bounded invariant set for all $\alpha$ and becomes a strong global attractor for $\alpha \geq 1/2$. TI - Blow-up in finite time for the dyadic model of the Navier-Stokes equations KW - blow-up KW - navier-stokes AU - Cheskidov, Alexey PY - 2006/Jan/04/ UR - http://arxiv.org/abs/math/0601074 ER -