Attractors of modified Kelvin — Voigt model with memory along fluid trajectories

Abstract

In the work we prove the existence of trajectory and global attractors for the modified Kelvin — Voigt model with memory along fluid trajectories. The proof is based on approximate-topological approach to study problems in the hydrodynamics.

Namely, first we introduce the needed functional spaces and give an operator interpretation of the considered problem. Then we pose an approximation problem and prove its solvability on a finite segment and on the semi-axis. Under certain conditions for the coefficients of the problem we establish exponential estimates of solutions, and these estimates are independent on the approximation parameter. After that, on the base of limit passage, we show the existence of a weak solution to the original problem on the semi--axis. Then we determine the trajectory space for the considered problem, show that the definition is well--defined and prove the existence theorem for minimal trajectory and global attractors.