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The attractors for the nonhomogeneous nonautonomous Navier–Stokes equations. (English) Zbl 1111.35042

The authors consider the inhomogeneous Navier-Stokes equations

$\begin{array}{cc}& {u}_{t}-\nu {\Delta }u+\left(u\nabla \right)u+\nabla p=f\hfill \\ & \text{div}\phantom{\rule{0.166667em}{0ex}}u=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}u=\varphi \phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },\phantom{\rule{1.em}{0ex}}{\Omega }\subseteq {ℝ}^{2}\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

on a bounded Lipschitz domain ${\Omega }$ in ${ℝ}^{2}$. One assumes

$f=f\left(x,t\right)\in {ℒ}_{\text{loc}}^{2}\left(\left(0,T\right),E\right),\phantom{\rule{1.em}{0ex}}\varphi \in {ℒ}^{2}\left(\partial {\Omega }\right)\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $E=\text{dom}\left({A}^{\frac{1}{4}}\right)$, with $A=-{P}_{{\Delta }}$ the Stokes operator associated with (1). The aim is to prove the existence of a global attractor for (1). To do so, the authors need several preparatory steps. First, using a suitable background flow $\psi$, eq. (1) is transformed into a new one, based on Dirichlet boundary conditions, i.e.:

${v}_{t}+\nu Av+B\left(v,v\right)+B\left(v,\psi \right)+B\left(\psi ,v\right)=P\left(f+\nu F\right)-B\left(\psi ,\psi \right)\phantom{\rule{2.em}{0ex}}\left(3\right)$

where $F$ is an additional force term induced by the background flow $\psi$. In order to prove the existence of an attractor for (3), the authors have to rely on work of V. V. Chepyzhov and M. I. Vishik [Am. Math. Soc. Colloq. Publ. 49, 363 p. (2002; Zbl 0986.35001)]; they introduce a number of notions and discuss their properties. Thus one has the notion of indexed process $\left\{{U}_{\sigma }\left(t,\tau \right)\mid t\ge \tau ,\phantom{\rule{4pt}{0ex}}\tau \in ℝ,\sigma \in {\Sigma }\right\}$ where ${\Sigma }$ is the index space (a metric space), $\sigma$ the symbol of the process and $\left\{{U}_{\sigma }\left(t,\tau \right)\right\}$ a family of mappings on a Banach space $E$ such that

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

In terms of this notion, the relevant topological concepts such as absorbing set, $\omega$-limit set, uniform attractor etc. are introduced, and some of their properties summarized. Criteria (Thms. 4.1, 4.2) for the existence of a uniform attractor are given. Finally, the index space is made precise: it is based on the translates $\left({T}_{h}f\right)\left(s\right)=f\left(h+s\right)$ induced by the exterior force $f$ in (1) resp. (3). In the main section 6 the existence of a uniform attractor in the sense of Chepyzhov and Vishik (loc. cit.) is proved. First, it is noted that existence of global solutions of (3) is guaranteed by a Galerkin method; for details the reader is referred to R. M. Brown, P. A. Perry, and Zh. Shen [Indiana Univ. Math. J. 49, 81–112 (2000; Zbl 0969.35105)] where a proof in a comparable situation is given. Then one proceeds to the proof of the main Theorem 6.1 which asserts the existence of a uniform attractor for (3). The proof involves lengthy estimates, based in part on the paper of Brown, Perry, Shen (loc. cit.). Theorem 6.2 finally asserts that if $f\left(x,s\right)$ is translation compact in $D\left({A}^{-\frac{1}{4}}\right)$ then the attractor in question is compact.

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