Intertial sets for a model of convection in a porous medium and for its time discretization

Abstract

The dynamical systems generated by a model of natural convection in a porous medium (1.1)–(1.2) and by its implicit time discretization (1.3)–(1.5) are considered. We study the dependence of the properties of these systems on the discretization parameter τ and on two physical constants Ra^* and γ. Inertial sets are constructed and are proved to be uniformly in γ close to each other in some natural sense when τ is small. Asymptotic estimates of the fractal dimension of the inertial sets are obtained and are shown to be uniform in τ and γ.

Keywords:

• Long time behavior
• inertial set
• porous media

¹Laboratoire de Mathématiques Applieuée Université Bordeaux I

²Steklov Mathematical Institute,St.Petersburg

¹Laboratoire de Mathématiques Applieuée Université Bordeaux I

²Steklov Mathematical Institute,St.Petersburg

Notes

¹Laboratoire de Mathématiques Applieuée Université Bordeaux I

²Steklov Mathematical Institute,St.Petersburg