Trajectory attractors method for dissipative partial differential equations with small parameter

Abstract

The purpose of this work is to study the limit behaviour of trajectory attractors for some equations and systems from mathematical physics depending on a small parameter when this small parameter approaches zero. The main attention is given to the cases when, for the limit equation, the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. The following problems are considered: approximation of the 3D Navier–Stokes system using the Leray α-model, homogenization of the complex Ginzburg–Landau equation in a domain with dense perforation, and zero viscosity limit of 2D Navier–Stokes system with Ekman friction. Methods. In this paper, the method of trajectory dynamical systems and trajectory attractors is used that is especially effective in the study of complicated partial differential equations for which the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. Results. For all problems under the consideration, we obtain the limit equations and prove the Hausdorff convergence for trajectory attractors of the initial equations to the trajectory attractors of the limit equations in the appropriate topology when the small parameter tends to zero. Conclusion. In the work, we demonstrate that the method of trajectory attractors is highly effective in the study of dissipative equations of mathematical physics with small parameter. We succeed to find the limit equations and to prove the convergence of trajectory attractors of the considered equations to the trajectory attractors of the limit (homogenized) equations in the corresponding topology as small parameter is vanishes.

Keywords

• global attractors
• trajectory attractors
• small parameter
• convergence of attractors