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

Abstract

In this paper, we aim to study the existence of global attractors for the time discretized modified three-dimensional (3D) Bénard systems. Using the backward implicit Euler scheme, we obtain the time discretization systems of 3D Bénard systems. Then, by the Galerkin method and the Brouwer fixed point theorem, we prove the existence of the solution to this time-discretized systems. On this basis, we proved the existence of the attractor by the compact embedding theorem of Sobolev. Finally, we discuss the limiting behavior of the solution as N tends to infinity.

Regularity and attractors for the three‐dimensional generalized Boussinesq system

“…Using the averaging technique that will give us the properties of the mean characteristics of the flow, they prove the global existence and uniqueness of the solutions and then obtain the existence of the global attractor. In Zhu [47], the existence of global attractors for the time discretized modified 3D Bénard systems is proved. In particular, in Yamazaki [32], the global regularity result of the 3D generalized Boussinesq equations with the same order of regularity for both $$$ u $$$ and $$$ \theta $$$ was studied.…”