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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 $$f(t)$$ 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 $$U_\sigma(t,\tau)$$ of nonlinar operators, acting on $$X$$, labeled by $$\sigma\in\Sigma$$ such that $U_\sigma(t,s) U_\sigma(s,\tau)= U_\sigma(t,\tau),\;t\geq s\geq \tau,\;U_\sigma(\tau, \tau)= \text{Id},\;\tau\in\mathbb{R},\;\sigma\in\Sigma.\tag{1}$ $$\Sigma$$ 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 $$C\in\mathbb{R}$$ and a bounded set $$B\subset X$$ there is $$T_0= T_0(\tau,B)\geq \tau$$ such that $\bigcup_{\sigma\in\Sigma} U_\sigma(t,\tau)B\subset B_0\quad \text{for }t\geq T_0.\tag{2}$ 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 $\partial_t u+\nu Au+ B(u,u)= f(t),\quad u(0)= u_0\tag{3}$ based on the Hilbert spaces \begin{aligned} H&= \{u\in L^2(\Omega)^2,\,\text{div}(u)= 0,\,u\cdot\vec n|_{\partial\Omega}= 0\},\text{ norm }|\;|,\\ V&= \{u\in H^1_0(\Omega)^2,\,\text{div}(u)= 0\},\text{ norm }\|\;\|. \end{aligned} The exterior force $$f(t)= \sigma(t)$$, $$t\in\mathbb{R}$$ is taken as symbol of the system (3), resp. of the induced process; one assumes $\sup_t \int^{t+1}_t |f(s)|^2\,ds< \infty.\tag{4}$ 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 $$f(t)$$, $$t\in\mathbb{R}$$ has an additional property called “normal” then the $$A_0$$ associated with $$f(t)$$, $$t\in\mathbb{R}$$ coincides with the uniform attractor $$A_b$$ associated with $$f(t+b)$$, $$t\in\mathbb{R}$$ , for any $$b\in\mathbb{R}$$. Further results of this type are obtained (Theorems 4.1, 4.2).

##### MSC:
 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 76D05 Navier-Stokes equations for incompressible viscous fluids
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