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Global attractors for small samples and germs of 3D Navier-Stokes equations. (English) Zbl 1075.35037

The author considers the Navier-Stokes equations

${u}_{t}=\nu {\Delta }u-\left(u\nabla \right)u+\nabla p+f,\phantom{\rule{4pt}{0ex}}\text{div}\left(u\right)=0,\phantom{\rule{4pt}{0ex}}u=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },\phantom{\rule{2.em}{0ex}}\left(1\right)$

with ${\Omega }$ smooth and bounded, and $f$ a constant exterior force. One imposes on (1) a standard Hilbert space setting:

${|u|}^{2}=\left(u,u\right)=\sum \int {u}_{j}^{2}\phantom{\rule{0.166667em}{0ex}}dx·$

$H$ is the ${L}^{2}$-closure of ${V}_{0}=\left\{u/u\in {C}_{0}{\left({\Omega }\right)}^{3},\text{div}\left(u\right)=0\right\}$, while $V$ is the closure of ${V}_{0}$ with respect to the norm

${\parallel u\parallel }^{2}=\sum \int {\left({\partial }_{j}u\right)}^{2}\phantom{\rule{0.166667em}{0ex}}dx,\phantom{\rule{1.em}{0ex}}{\partial }_{j}={\partial }_{{x}_{j}}·$

A weak solution of (1) is an element $u$ in ${L}^{\infty }\left(0,\infty ;H\right)\cap {L}^{2}\left(0,T;V\right)$ (for all $T$) subject to three classical conditions, one requiring that $u$ satisfies a standard variational form of (1). The set $W$ of weak solutions is considered as a metric space under the norm

${\left[u\right]}^{2}={\int }_{0}^{\infty }{|u\left(t\right)|}^{2}{e}^{-t}\phantom{\rule{0.166667em}{0ex}}dt·$

It was proved by G. R. Sell [J. Dyn. Differ. Eq. 8, No. 1, 1–33 (1996; Zbl 0855.35100)] that the semiflow ${S}_{t}$ on $W$ given by

$\left({S}_{t}u\right)\left(x\right)=u\left(t+s\right),\phantom{\rule{1.em}{0ex}}s\ge 0$

admits a global attractor $A$. The author now takes a generalization of this situation given by J. M. Ball [J. Nonlinear Sci. 7, No. 5, 475–502 (1997; Zbl 0903.58020)] as starting point who introduced the notion of generalized semiflow $G$ on a metric space $X$. Since this notion is not directly applicable to the space $W$, the author defines a notion of $\epsilon$-samples as follows. With $u|\epsilon$ the restriction of $u\in {L}^{\infty }\left(0,\infty ;H\right)$ to $\left[0,\epsilon \right]$ one sets:

${W}_{\epsilon }=\left\{u|\epsilon \mid u\in W\right\}$

and with $u\in W$ one associates ${\varphi }_{u}^{\epsilon }:\left[0,\infty \right]\to W$ as follows:

${\varphi }_{u}^{\epsilon }\left(t\right)={u}^{t}|\epsilon ,\phantom{\rule{1.em}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}{u}^{t}\left(s\right)=u\left(t+s\right),\phantom{\rule{1.em}{0ex}}s\ge 0·$

One then sets:

${G}_{\epsilon }=\left\{{\varphi }_{u}^{\epsilon }\mid u\in W\right\}·$

The main results then are:

(I) ${G}_{\epsilon }$ is a generalized semiflow on ${W}_{\epsilon }$,

(II) with $A$ the attractor for ${S}_{t}$ and ${A}_{\epsilon }=\left\{u\mid \epsilon \mid u\in A\right\}$, ${A}_{\epsilon }$ is a global attractor for ${G}_{\epsilon }$.

The author extends the above theory to a familiy of objects called “germs”, which are a kind of infinitesimal $\epsilon$-samples $\left(\epsilon =0\right)$.

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