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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 Ω:=×[-a,a] 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:={u[L 2 (Ω)] 2 ,divu=0}

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:={uH 2 (Ω),divu=0,u| Ω =0}·

MSC:
35B41Attractors (PDE)
35Q35PDEs in connection with fluid mechanics
37L30Attractors and their dimensions, Lyapunov exponents