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Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces. (English) Zbl 1083.35094

The authors prove he existence of attractors for the 2d-Navier-Stokes equations with an exterior force $f\left(t\right)$ by using the theory of processes by Babin, Vishik and others. A process is based on a Banach space $X$, an index set ${\Sigma }$ (itself a topological vector space). A process is a collection ${U}_{\sigma }\left(t,\tau \right)$ of nonlinar operators, acting on $X$, labeled by $\sigma \in {\Sigma }$ such that

${U}_{\sigma }\left(t,s\right){U}_{\sigma }\left(s,\tau \right)={U}_{\sigma }\left(t,\tau \right),\phantom{\rule{4pt}{0ex}}t\ge s\ge \tau ,\phantom{\rule{4pt}{0ex}}{U}_{\sigma }\left(\tau ,\tau \right)=\text{Id},\phantom{\rule{4pt}{0ex}}\tau \in ℝ,\phantom{\rule{4pt}{0ex}}\sigma \in {\Sigma }·\phantom{\rule{2.em}{0ex}}\left(1\right)$

${\Sigma }$ is called the symbol space, $\sigma$ a symbol. Concepts such as uniform attractor, uniform absorbing set etc. are now introduced. E.g., ${B}_{0}\subset X$ is uniformly absorbing if given $C\in ℝ$ and a bounded set $B\subset X$ there is ${T}_{0}={T}_{0}\left(\tau ,B\right)\ge \tau$ such that

$\bigcup _{\sigma \in {\Sigma }}{U}_{\sigma }\left(t,\tau \right)B\subset {B}_{0}\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}t\ge {T}_{0}·\phantom{\rule{2.em}{0ex}}\left(2\right)$

The authors now prove a number of preparatory lemmas concerning properties of processes. These results are then applied to the 2d-Navier-Stokes equation on a smooth bounded domain ${\Omega }$. To this end this equation is put into standard abstract form

${\partial }_{t}u+\nu Au+B\left(u,u\right)=f\left(t\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)={u}_{0}\phantom{\rule{2.em}{0ex}}\left(3\right)$

based on the Hilbert spaces

$\begin{array}{cc}\hfill H& =\left\{u\in {L}^{2}{\left({\Omega }\right)}^{2},\phantom{\rule{0.166667em}{0ex}}\text{div}\left(u\right)=0,\phantom{\rule{0.166667em}{0ex}}u·\stackrel{\to }{n}{|}_{\partial {\Omega }}=0\right\},\phantom{\rule{4.pt}{0ex}}\text{norm}\phantom{\rule{4.pt}{0ex}}|\phantom{\rule{4pt}{0ex}}|,\hfill \\ \hfill V& =\left\{u\in {H}_{0}^{1}{\left({\Omega }\right)}^{2},\phantom{\rule{0.166667em}{0ex}}\text{div}\left(u\right)=0\right\},\phantom{\rule{4.pt}{0ex}}\text{norm}\phantom{\rule{4.pt}{0ex}}\parallel \phantom{\rule{4pt}{0ex}}\parallel ·\hfill \end{array}$

The exterior force $f\left(t\right)=\sigma \left(t\right)$, $t\in ℝ$ is taken as symbol of the system (3), resp. of the induced process; one assumes

$\underset{t}{sup}{\int }_{t}^{t+1}{|f\left(s\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}ds<\infty ·\phantom{\rule{2.em}{0ex}}\left(4\right)$

After recalling global existence and uniqueness of solutions of (3) the authors proceed to prove the existence of a uniform attractor ${A}_{0}$ of (3) and investigate its properties. The relevant Theorem 3.3 states among others (expressed somewhat losely) that if $f\left(t\right)$, $t\in ℝ$ has an additional property called “normal” then the ${A}_{0}$ associated with $f\left(t\right)$, $t\in ℝ$ coincides with the uniform attractor ${A}_{b}$ associated with $f\left(t+b\right)$, $t\in ℝ$ , for any $b\in ℝ$. Further results of this type are obtained (Theorems 4.1, 4.2).

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 35B40 Asymptotic behavior of solutions of PDE 35B41 Attractors (PDE) 37L30 Attractors and their dimensions, Lyapunov exponents 76D05 Navier-Stokes equations (fluid dynamics)