Finite-dimensional attractors for the Kirchhoff models

Abstract

The paper studies the existence of the finite-dimensional global attractor and exponential attractor for the dynamical system associated with the Kirchhoff models arising in elasto-plastic flow utt−div{|∇u|m−1∇u}−Δut+Δ2u+h(ut)+g(u)=f(x). By using the method of ℓ-trajectories and the operator technique, it proves that under subcritical case, 1≤m<N+2(N−2)+, the above-mentioned dynamical system possesses in different phase spaces a finite-dimensional (weak) global attractor and a weak exponential attractor, respectively. For application, the fact shows that for the concerned elasto-plastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the fractal 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.