Global attractors and their Hausdorff dimensions for a class of Kirchhoff models

Abstract

This paper studies the existence, regularity, and Hausdorff dimensions of global attractors for a class of Kirchhoff models arising in elastoplastic flow utt−div{|∇u|m−1∇u}−Δut+Δ2u+h(ut)+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above-mentioned models possesses in phase space X a global attractor which has further regularity in Xσ1(↪↪X) and has finite Hausdorff dimension. For application, the fact shows that for the concerned elastoplastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.