Navier-Stokes Equations in Thin 3D Domains III: Existence of a Global Attractor

Abstract

Over 50 years ago, Leray (1933, 1934a, 1934b) published his pioneering works on the Navier-Stokes equations which led to the modern mathematical theory of fluid dynamics. These equations describe the time evolution of solutions of mathematical models of viscous incompressible fluid flows. Since the solutions of these equations depend on both space and time, one is especially interested in the phenomenon of the time evolution of the spatial variations of the solutions. This phenomenon, which is described with more precision later, is referred to as the regularity of solutions and it is one of our concerns here. During the last few years, much attention has also been focused on the study of attractors for the Navier-Stokes equations. This is related to a new insight in turbulence, which relies two theories of turbulence, the conventional theory of turbulence and the dynamical system approach (see, for instance, Babin and Vishik (1983), Constantin and Foias (1985), Constantin, Foias and Temam (1985), Constantin, Foias, Manley and Temam (1985), Foias and Temam (1987)).