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${H}^{2}$-compact attractor for a non-Newtonian system in two-dimensional unbounded domains. (English) Zbl 1073.35044

The authors continue their study of the long-time behavior of the bipolar viscous non-Newtonian fluid in two-dimensional infinite strip ${\Omega }:=ℝ×\left[-a,a\right]$ started in [Y. Li and C. Zhao, Acta Anal. Funct. Appl. 4, No. 4, 343–349 (2002; Zbl 1053.35117)]. In the previous paper the existence of a global attractor for that problem in the phase space

$H:=\left\{u\in {\left[{L}^{2}\left({\Omega }\right)\right]}^{2},\phantom{\rule{4pt}{0ex}}divu=0\right\}$

were established [see also F. Bloom and W. Hao, Nonlinear Anal., Theory Methods Appl. 43, No. 6, 743–766 (2001; Zbl 0989.76003)], where the analogous result were established for the external forces belonging to the appropriate weighted Sobolev spaces. The main result of the present paper is the existence of a compact global attractor in a more regular phase space

$V:=\left\{u\in {H}^{2}\left({\Omega }\right),\phantom{\rule{4pt}{0ex}}divu=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u{|}_{\partial {\Omega }}=0\right\}·$

##### MSC:
 35B41 Attractors (PDE) 35Q35 PDEs in connection with fluid mechanics 37L30 Attractors and their dimensions, Lyapunov exponents