On an Iterative Method for Approximate Solutions of a Generalized Boussinesq Model

Abstract

Abstract. An iterative method is proposed for finding approximate solutions of an initial and boundary value problem for a nonstationary generalized Boussinesq model for thermally driven convection of fluids with temperature dependent viscosity and thermal conductivity. Under certain conditions, it is proved that such approximate solutions converge to a solution of the original problem; moreover, convergence-rate bounds for the constructed approximate solutions are also obtained.

Global regularity for a two‐dimensional nonlinear Boussinesq system

“…On the one hand, it is the Boussinesq system () with full nonlinear dissipation ( ν = ν ( θ ) > 0, κ = κ ( θ ) > 0). Boldrini, Climent‐Ezquerra, Rojas‐Medar, and Rojas‐Medar proposed an iterative method for finding approximate solutions. In , Boldrini, Fernández‐Cara, and Rojas‐Medar obtained the existence of an optimal solution.…”

“…Moreover, it will be very hard to get the high‐order estimates because we need a higher order estimate when we have obtained a high‐order estimate (e.g., when we estimate ∥∇ u ∥, we need to estimate ∥Δ u ∥); it is an endless iteration. However, in the case of the full dissipation ( ν = ν ( θ ) > 0, κ = κ ( θ ) > 0), this problem will not happen (e.g., ), because the higher order terms on the right can be controlled by the left terms. In order to overcome them, we are going to utilize the vanishing diffusivity method to prove the existence of global quasi‐strong solutions and get some higher order uniform estimates, and furthermore prove the global well‐posedness of the Boussinesq system () and ().…”

Exact controllability of Galerkin’s approximations of micropolar fluids

“…The results given in our work can be applied to different models of fluid mechanics. Some of these models, such as generalized Boussinesq systems (see [2]) and equations describing liquid crystals (see [17]) are intensively studied. This paper is organized as follows: In Section 2, we state the basic notations.…”