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Zbl 1083.35094
Lu, Songsong; Wu, Hongqing; Zhong, Chengkui
Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces.
(English)
[J] Discrete Contin. Dyn. Syst. 13, No. 3, 701-719 (2005). ISSN 1078-0947; ISSN 1553-5231/e

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 $$U_\sigma(t,s) U_\sigma(s,\tau)= U_\sigma(t,\tau),\ t\ge s\ge \tau,\ U_\sigma(\tau, \tau)= \text{Id},\ \tau\in\bbfR,\ \sigma\in\Sigma.\tag1$$ $\Sigma$ 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 $C\in\bbfR$ and a bounded set $B\subset X$ there is $T_0= T_0(\tau,B)\ge \tau$ such that $$\bigcup_{\sigma\in\Sigma} U_\sigma(t,\tau)B\subset B_0\quad \text{for }t\ge T_0.\tag2$$ 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 $$\partial_t u+\nu Au+ B(u,u)= f(t),\quad u(0)= u_0\tag3$$ based on the Hilbert spaces \align 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 }\Vert\ \Vert. \endalign The exterior force $f(t)= \sigma(t)$, $t\in\bbfR$ is taken as symbol of the system (3), resp. of the induced process; one assumes $$\sup_t \int^{t+1}_t |f(s)|^2\,ds< \infty.\tag4$$ 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 $f(t)$, $t\in\bbfR$ has an additional property called normal'' then the $A_0$ associated with $f(t)$, $t\in\bbfR$ coincides with the uniform attractor $A_b$ associated with $f(t+b)$, $t\in\bbfR$ , for any $b\in\bbfR$. Further results of this type are obtained (Theorems 4.1, 4.2).
[Bruno Scarpellini (Basel)]
MSC 2000:
*35Q30 Stokes and Navier-Stokes equations
35B40 Asymptotic behavior of solutions of PDE
35B41 Attractors
37L30 Attractors and their dimensions
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: existence of attractors; Navier-Stokes equations; uniform attractor; uniform absorbing set; global existence and uniqueness of solutions

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