Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$

@article{Chepyzhov2015StrongTA,
  title={Strong trajectory and global \$\mathbf\{W^\{1,p\}\}\$-attractors for the damped-driven Euler system in \$\mathbb R^2\$},
  author={V. Chepyzhov and A. Ilyin and S. Zelik},
  journal={arXiv: Analysis of PDEs},
  year={2015}
}
We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space $H^1$. A similar result on the strong attraction holds in the spaces $H^1\cap\{u:\ \|\mathrm{curl} u\|_{L^p}<\infty\}$ for $p\ge2$. 
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