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Attractors for 2D-Navier-Stokes models with delays. (English) Zbl 1068.35088

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

$\begin{array}{cc}\hfill {\partial }_{t}u=\nu {\Delta }u-\nabla p-\left(u\nabla \right)u+f-g\left(t,{u}_{t}\right)\phantom{\rule{1.em}{0ex}}& \text{in}\phantom{\rule{4.pt}{0ex}}\left(\tau ,\infty \right)×{\Omega },\hfill \\ \hfill \text{div}\left(u\right)=0,\phantom{\rule{4pt}{0ex}}u=0\phantom{\rule{1.em}{0ex}}& \text{on}\phantom{\rule{4.pt}{0ex}}\left(\tau ,\infty \right)×\partial {\Omega },\phantom{\rule{4pt}{0ex}}u\left(\tau ,x\right)={u}_{0}\left(x\right),\hfill \\ \hfill u\left(t,x\right)=\phi \left(t-\tau ,x\right)\phantom{\rule{1.em}{0ex}}& \text{for}\phantom{\rule{4.pt}{0ex}}t\in \left(\tau -h,\tau \right),\phantom{\rule{4pt}{0ex}}x\in {\Omega }·\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

Here, ${u}_{t}\left(s\right)=u\left(t+s\right)$, $s\in \left(-h,0\right)$; the delay is encoded in $\phi$, $g$. ${\Omega }\subset {ℝ}^{2}$ 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 ${V}_{0}=\left\{u\in {C}_{0}^{\infty }\left({\Omega }\right)$, $\text{div}\left(u\right)=0\right\}$, 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 ${C}_{H}={C}^{0}\left(\left[-h,0\right],H\right)$ etc. The delay is taken into account by $g\left(t,\xi \right)$ which is a mapping from $ℝ×{C}_{H}$ to ${L}^{2}{\left({\Omega }\right)}^{2}$. The function $g\left(t,\xi \right)$ is subject to a series of assumption involving measurability and two kinds of Lipschitz conditions. System (1) is then put into the abstract form

${\partial }_{t}u+\nu Au+Bu=f+g\left(t,{u}_{t}\right),\phantom{\rule{4pt}{0ex}}u\left(0\right)={u}_{0}\phantom{\rule{2.em}{0ex}}\left(2\right)$

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 $\left\{A\left(t\right),\phantom{\rule{0.166667em}{0ex}}t\in ℝ\right\}$ called pullback attractor, related to a family $U\left(t,s\right)$ 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