Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function
type: Item
instance of: scholarly article
title: Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function
zbMATH Open document ID: 0843.35074
author: Yoshiyuki Kagei
published in: Hiroshima Mathematical Journal
publication date: 1996-08-22T00:00:00Z
review text: The author studies the Bénard convection problem with dissipative heating in a domain \(\Omega= (0, \alpha)\times (0, 1)\subset \mathbb{R}^2\), i.e. \[ u_t- \nu\Delta u+ (u\cdot \nabla) u+ \nabla p= e_2 f(\theta),\quad\text{div } u= 0\quad\text{in } \Omega\times (0, \infty), \] \[ \theta_t- \kappa\Delta\theta+ u\cdot \nabla\theta- e_2\cdot u= {\eta\nu\over 2} D(u): D(u)\quad\text{in } \Omega\times (0, \infty), \] together with the boundary conditions \(u\), \(\theta= 0\) on \(x_2= 0, 1\) while the solution is required to be periodic in \(x_1\). The dissipation function is given by \[ D(u)= \sum^2_{i,k= 1} \Biggl({\partial u^i\over \partial x_k}+ {\partial u^k\over \partial x_i}\Biggr)^2 \] and \(f\) is a smooth function satisfying either \(f(\theta)= \theta\) or \(\sup|f, f', f''|< \infty\). Using energy estimates, the author establishes existence and uniqueness of solutions as well as their continuous dependence on the initial data. After that he proves the existence of a global attractor and gives an estimate for its Hausdorff dimension involving the relevant physical parameters.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76R10
zbMATH DE Number: 811373
zbMATH Keywords: Bénard convection problem
zbMATH Keywords: dissipation function
zbMATH Keywords: existence of a global attractor
zbMATH Keywords: Hausdorff dimension
reviewed by: Klaus Deckelnick
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 811373