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

TL;DR

This paper proves the existence of strong global and trajectory attractors for the damped-driven 2D Euler equations in the plane, in various Sobolev spaces, demonstrating long-term stability of solutions.

Contribution

It establishes the existence of strong attractors in Sobolev spaces for the damped-driven Euler system, including cases with finite enstrophy and higher integrability.

Findings

Solutions satisfy energy and enstrophy equality.

Existence of strong global attractors in H^1.

Attraction results extend to spaces with finite curl in L^p for p ≥ 2.

Abstract

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 $p\ge2$.