Attractors theory for autonomous systems of hydrodynamics and its application to Bingham model of fluid motion

Abstract

In the study of solutions behavior of various mathematical physics equations their limit state when time tends to infinity (so-called attractors) is of particular interest. At the works of russian mathematicians M.I. Vishik and V.V. Chepyzhov and american mathematician G. Sell a new approach for the attractors study based on the consideration of trajectory spaces and trajectory attractors of the corresponding equations was proposed. However, for a large number of hydrodynamics equations this theory could not be applied because of the conditions of the translational invariance and of the closure of a trajectory space. In the introduction of this article a brief summary of the attractors theory for autonomous systems based on the concept of the trajectory space without the assumption of its translational invariance is presented. As an application of this theory, the existence of the global attractor for autonomous model of the Bingham medium motion in the case of periodic spatial variables in proved.