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Weak and strong attractors for the 3D Navier-Stokes system. (English) Zbl 1131.35056

The authors study the asymptotic behaviour of weak solutions of the Navier-Stokes equations

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

considered on a smooth bounded domain ${\Omega }\subseteq {ℝ}^{3}$, and subject to homogeneous Dirichlet boundary conditions; here $A=-{P}_{{\Delta }}$ is the Stokes operator, while $B\left(u,u\right)=P\left(u\nabla \right)u$ is the nonlinearity. Eq. (1) is studied in an ${L}^{2}$-setting based on the familiar spaces $H$, $V$ [R. Temam, Navier-Stokes equations. Theory and numerical analysis. Amsterdam etc.: North-Holland (1979; Zbl 0426.35003)]. Following these lines, a standard definition of weak solution is given. Starting point is an unproved assumption (H) imposed on the exterior force $f\left(t\right)$:

(H) Let $f\in {L}_{\text{for}}^{2}\left(\left[\tau ,\infty \right),H\right)$; there exists a function $F\left(\alpha ,\beta ,\gamma \right)$, continuous and nondecreasing in $\alpha$ and nonincreasing in $\beta$, such that for any ${u}_{\tau }\in V$ there exists a globally defined weak solution $u\left(\phantom{\rule{4pt}{0ex}}\right)$ such that $u\left(\tau \right)={u}_{\tau }$ and

${\parallel u\left(t\right)\parallel }_{{L}^{4}{\left({\Omega }\right)}^{3}}\le F\left(\parallel {u}_{\tau }{\parallel }_{V},\tau ,T\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}T>\tau \phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}t\in \left[\tau ,T\right]·\phantom{\rule{2.em}{0ex}}\left(2\right)$

It then follows that the weak solutions $u\left(\phantom{\rule{4pt}{0ex}}\right)$ provided by assumption (H) satisfy

${V}_{\tau }\left(u\left(t\right)\right)\le {V}_{\tau }\left(u\left(s\right)\right),\phantom{\rule{1.em}{0ex}}t\ge s\ge \tau \phantom{\rule{2.em}{0ex}}\left(3\right)$

where

${V}_{\tau }\left(u\left(t\right)\right)=\frac{1}{2}{\parallel u\left(t\right)\parallel }^{2}+\nu {\int }_{\tau }^{t}{\parallel \left(\nabla u\right)\left(r\right)\parallel }^{2}\phantom{\rule{0.166667em}{0ex}}dr-{\int }_{\tau }^{t}\left(f\left(r\right),u\left(r\right)\right)\phantom{\rule{0.166667em}{0ex}}dr,$

and are unique in the class of solutions which satisfy (3). The authors then prove a series of theorems, all expressing properties of weak solutions provided by (H). Based on these preparations, the authors then proceed to study a class of attractors associated with these weak solutions. They set $\tau =0$ in (1)–(3), assume that $f\in H$ in (1) is autonomous and define a set valued mapping $G$ from ${ℝ}_{+}×H$ into the set $P\left(H\right)$ of all nonempty subsets of $H$ as follows:

(4) $G\left(t,{u}_{0}\right)$ is the set of all $u\left(t\right)$, where $u\left(\phantom{\rule{4pt}{0ex}}\right)$ is a globally defined weak solution subject to (3), and with $u\left(0\right)={u}_{0}·$ It is proved that $G$ is a strict multivalued semiflow, i.e.

$G\left(0,{u}_{0}\right)={u}_{0}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}G\left({t}_{1},G\left({t}_{2},{u}_{0}\right)\right)=G\left({t}_{1}+{t}_{2},{u}_{0}\right)·$

For such semiflows the notion of global attractor can be defined along established lines. The authors prove that the semiflow $G$ has a global attractor $𝒜$, which is minimal. The case of nonautonomous $f\left(t\right)$ is more involved and requires the introduction of multivalued processes in the sense of A. V. Babin and M. I. Vishik [Attractors of evolution equations. Moscow: Nauka (1989; Zbl 0804.58003)]. The authors prove the existence of a global attractor in this more general situation. The paper concludes with a theory of attractors for multivalued processes on topological spaces, extending results obtained by V. S. Melnik and J. Valero [Set-Valued Anal. 8, No. 4, 375–403 (2000; Zbl 1063.35040)] for metric spaces.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 37L30 Attractors and their dimensions, Lyapunov exponents 35B41 Attractors (PDE) 35K55 Nonlinear parabolic equations 37B25 Lyapunov functions and stability; attractors, repellers 58C06 Set-valued and function-space valued mappings on manifolds 35B40 Asymptotic behavior of solutions of PDE