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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 \(\Omega:=\mathbb R\times[-a,a]\) 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 \[ H:=\{u\in [L^2(\Omega)]^2,\;\operatorname {div}u=0\} \] 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 \[ V:=\{u\in H^2(\Omega),\;\operatorname {div}u=0,\;\;u\big| _{\partial\Omega}=0\}. \]

MSC:

35B41 Attractors
35Q35 PDEs in connection with fluid mechanics
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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