Averaging of trajectory attractors of evolution equations with rapidly oscillating terms

Abstract

Evolution equations containing rapidly oscillating terms with respect to the spatial variables or the time variable are considered. The trajectory attractors of these equations are proved to approach the trajectory attractors of the equations whose terms are the averages of the corresponding terms of the original equations. The corresponding Cauchy problems are not assumed here to be uniquely soluble. At the same time if the Cauchy problems for the equations under consideration are uniquely soluble, then they generate semigroups having global attractors. These global attractors also converge to the global attractors of the averaged equations in the corresponding spaces. These results are applied to the following equations and systems of mathematical physics: the 3D and 2D Navier-Stokes systems with rapidly oscillating external forces, reaction-diffusion systems, the complex Ginzburg-Landau equation, the generalized Chafee-Infante equation, and dissipative hyperbolic equations with rapidly oscillating terms and coefficients.