Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Cited in: Zbl 1187.35016 Zbl 1195.35062 Zbl 1170.35025 Zbl 1149.35328

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]