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Existence and regularity of pullback attractors for an incompressible non-Newtonian fluid with delays

Author(s): Caidi Zhao; Shengfan Zhou; Yongsheng Li
Journal: Quart. Appl. Math. 67 (2009), 503-540.
MSC (2000): Primary 35B41, 35Q35, 76D03
Posted: May 6, 2009
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Abstract: This paper studies an incompressible non-Newtonian fluid with delays in two-dimensional bounded domains. We first prove the existence and uniqueness of solutions. Then we establish the existence of pullback attractors $ \{\mathscr{A}_{_{\mathcal{C}_{_H}}}(t)\}_{t\in \mathbb{R}}$ (has $ L^2$-regularity), $ \{\mathscr{A}_{_{\mathcal{C}_{_W}}}(t)\}_{t\in \mathbb{R}}$ (has $ H^2$-regularity), and $ \{\mathscr{A}_{_{E^2_{_H}}}(t)\}_{t\in \mathbb{R}}$ (has $ L^2$-regularity), $ \{\mathscr{A}_{_{E^2_{_W}}}(t)\}_{t\in \mathbb{R}}$ (has $ H^2$-regularity) corresponding to two different processes associated to the fluid, respectively. Meanwhile, we verify the regularity of the pullback attractors by proving

$\displaystyle \mathscr{A}_{_{\mathcal{C}_{_H}}}(t) =\mathscr{A}_{_{\mathcal{C}_... ..._{_{E^2_{_H}}}(t)=\mathscr{A}_{_{E^2_{_W}}}(t), \quad \forall t\in \mathbb{R}, $

and

$\displaystyle \mathscr{A}_{_{E^2_{_H}}}(t) =J(\mathscr{A}_{_{\mathcal{C}_{_H}}}... ...hcal{C}_{_W}}}(t)) =\mathscr{A}_{_{E^2_{_W}}}(t),\quad \forall t\in\mathbb{R}, $

where $ J$ is a linear operator. By the regularity we reveal the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data. This effect implies, in the case of delays, that the regularity of the fluid in its history state does not play an important role on the regularity of its eventual state. Finally, we give some remarks.

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Additional Information:

Caidi Zhao
Affiliation: Department of Mathematics, Wenzhou University, Wenzhou 325035, People's Republic of China
Email: zhaocaidi@yahoo.com.cn

Shengfan Zhou
Affiliation: Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, People's Republic of China

Yongsheng Li
Affiliation: Department of Applied Mathematics, South China University of Technology, Guangdong, 510640, People's Republic of China

PII: S0033-569X-09-01146-2
Keywords: Pullback attractor, evolution equations, incompressible non-Newtonian fluid, pullback asymptotic smoothing effect
Received by editor(s): February 20, 2008
Posted: May 6, 2009
Additional Notes: The work is supported by the NSF of China under Grant Numbers 10826091, 10771139, 10471047, and 10771074, the NSF of Zhejiang Province under Grant Number Y6080077 and the Innovation Program of Shanghai Municipal Education Commission under Grant 08ZZ70
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