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# Dynamics of Partial Differential Equations

## Volume 10 (2013)

### Number 3

### Dynamics of non-autonomous equations of non-Newtonian fluid on 2D unbounded domains

Pages: 283 – 312

DOI: http://dx.doi.org/10.4310/DPDE.2013.v10.n3.a4

#### Author

#### Abstract

This paper studies the asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid on two-dimensional unbounded domains. We first prove the existences of the $L^2$-regularity uniform attractors $\mathcal{A}^H_{\mathcal{H}(g_0)}$ and $H^2$-regularity uniform attractor $\mathcal{A}^V_{\mathcal{H}(g_0)}$, respectively. Then we establish the regularity of the uniform attractors by showing$$\mathcal{A}^H_{\mathcal{H}(g_0)}=\mathcal{A}^V_{\mathcal{H}(g_0)},$$which implies the uniform (with respect to the external forces) asymptotic smoothing effect of the non-autonomous fluid in the sense that the solutions become eventually more regular than the initial data.

#### Keywords

non-autonomous non-Newtonian fluids, uniform attractor, uniform asymptotic smoothing effect

#### 2010 Mathematics Subject Classification

Primary 35B41, 35Q35. Secondary 76D03.

Published 11 October 2013