Attractor of a semi-discrete Benjamin–Bona–Mahony equation on R¹

Abstract

This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize the Benjamin-Bona-Mahony equation on R 1 . Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H 1 (R 1 ). Then we prove that this system possesses a global attractor Aτ in H 1 (R 1 ). In addition, we show that the global attractor Aτ is regular, i.e., Aτ is actually included, bounded and compact in H 2 (R 1 ). Finally, we estimate the finite fractal dimensions of Aτ .

Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system

“…This bound is independent of the time step and, although crude, it is explicit in terms of the physical parameters of the model. In contrast, only a few authors have obtained upper bounds independent of the discretization parameters for the dimensions of attractors, by using more specific methods [41,42,44].…”

Global attractor for the time discretized modified three-dimensional Bénard systems

“…In [5], the modified 3D Navier-Stokes equations were discretized on the time by finite difference method, then the existence of the global attractor was proved. In the literature [15], the Benjamin-Bona-Mahony equation was discretized on the time by the Crank-Nicolson scheme. Then, using the Galerkin method and the Brouwer fixed point theorem, authors proved that the existence of the solution to this time discretized system.…”

“…The main purpose of this paper is to investigate the long time dynamical behavior of the solution of the discretized, modified 3D Bénard system (1.1)-(1.2) by the idea in [5,15].…”

On the Caginalp for a Conserve Phase-Field with a Polynomial Potentiel of Order 2<i>p</i> - 1

“…is the conductive thermal displacement. Noting that T t α ∂ = ∂ , we finally deduce from (33) and (36)-(37) the following variant of the Caginalp phase-field system (see [17]):…”

“…Our aim in this paper is to study the existence and uniqueness of solution of (17)-(39). We consider here only one type of boundary condition, namely, Dirichlet (see [31] [32] [33]).…”