*(English)*Zbl 1068.35088

The authors consider the Navier-Stokes equations with delay in 2D. The equation in question is

Here, ${u}_{t}\left(s\right)=u(t+s)$, $s\in (-h,0)$; the delay is encoded in $\phi $, $g$. ${\Omega}\subset {\mathbb{R}}^{2}$ is a smooth bounded domain. In order to cast (1) into an abstract form, a standard ${L}^{2}$-setting is imposed on (1). E.g., $H$ is the ${L}^{2}$-closure of ${V}_{0}=\{u\in {C}_{0}^{\infty}\left({\Omega}\right)$, $\text{div}\left(u\right)=0\}$, while $V$ is the closure of ${V}_{0}$ with respect to the norm $\parallel \nabla u\parallel $. In order to take care of the delay, spaces ${C}_{H}$, ${C}_{V}$, ${L}_{H}^{2}$ are introduced, where ${C}_{H}={C}^{0}([-h,0],H)$ etc. The delay is taken into account by $g(t,\xi )$ which is a mapping from $\mathbb{R}\times {C}_{H}$ to ${L}^{2}{\left({\Omega}\right)}^{2}$. The function $g(t,\xi )$ is subject to a series of assumption involving measurability and two kinds of Lipschitz conditions. System (1) is then put into the abstract form

along conventional lines. The authors first recall a result obtained in an earlier paper, stating that (under some assumptions) (2) admits a unique global solution in some suitable function space. The authors then proceed to show that (2) admits a global attractor.

In this context, the usual notion of attractor does not suffice; it has to be replaced by a new notion, i.e. by a family $\left\{A\right(t),\phantom{\rule{0.166667em}{0ex}}t\in \mathbb{R}\}$ called pullback attractor, related to a family $U(t,s)$ of evolution operators associated with (2). After proving the existence of an absorbing set, the authors then show that under suitable assumptions on the data, a unique, uniformly bounded pullback attractor for (2) exists. The paper concludes with an application to the case where (2) is supplied by a forcing term.

##### MSC:

35Q30 | Stokes and Navier-Stokes equations |

35R10 | Partial functional-differential equations |

47H20 | Semigroups of nonlinear operators |

37L30 | Attractors and their dimensions, Lyapunov exponents |