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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 $$\aligned & u_t-\nu\Delta u+(u\nabla)u+\nabla p=f\\ & \text{div }u=0\text{ on }\Omega,\quad u=\varphi\text{ on }\partial\Omega, \quad \Omega\subseteq \bbfR^2\endaligned\tag1$$ on a bounded Lipschitz domain $\Omega$ in $\Bbb R^2$. One assumes $$f=f(x,t)\in{\cal L}^2_{\text{loc}}((0,T),E),\quad \varphi\in{\cal L}^2(\partial\Omega)\tag2$$ where $E=\text{dom}(A^{\frac14})$, 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.: $$v_t+\nu Av+B(v,v)+B(v,\psi)+B(\psi,v)=P(f+\nu F)-B(\psi,\psi)\tag3$$ 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 {\it V. V. Chepyzhov} and {\it 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 $\{U_\sigma(t,\tau)\mid t\ge \tau,\ \tau\in\Bbb R, \sigma\in\Sigma\}$ where $\Sigma$ is the index space (a metric space), $\sigma$ the symbol of the process and $\{U_\sigma(t,\tau)\}$ a family of mappings on a Banach space $E$ such that $$U(t,s)U(s,\tau)=U(t,\tau),\quad U(\tau,\tau)=\text{Id},\quad t\ge s\ge \tau,\quad \tau\in\Bbb R.$$ 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 $(T_hf)(s)=f(h+s)$ 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 {\it R. M. Brown, P. A. Perry}, and {\it 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 $f(x,s)$ is translation compact in $D(A^{-\frac14})$ then the attractor in question is compact.

MSC:
35Q30Stokes and Navier-Stokes equations
37L30Attractors and their dimensions, Lyapunov exponents
76D05Navier-Stokes equations (fluid dynamics)
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References:
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