Global existence and exponential stability of convection
type: Item
instance of: scholarly article
title: Global existence and exponential stability of convection
zbMATH Open document ID: 0862.35089
author: Toshiaki Hishida
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-05-26T00:00:00Z
review text: The interior convection in the Boussinesq approximation is studied for a bounded domain \(\Omega\subset \mathbb{R}^3\), occupied by an incompressible, viscous fluid, with a smooth boundary \(\Gamma\). It is studied the gravitational convection when the fluid is heated at a part \(\Gamma_0\subset\Gamma\) of the boundary. The movement is governed by the system of equations: \[ \partial_tv+ v\cdot\nabla v= \nu\Delta v+g_0(\nabla\psi)(1-\eta(\theta-T_0))- \nabla p\quad\text{for } x\in\Omega,\quad t>0; \] \[ \nabla v=0\quad\text{for }x\in\Omega,\quad t>0;\quad \partial_t\theta+v\nabla\theta= h\Delta\theta\quad\text{for }x\in\Omega,\quad t>0, \] in the boundary conditions: \(v=0\), \(\theta=T_w(x)\neq\text{const.}\) for \(x\in\Gamma\), \(t>0\). The following notations were used: \(v(x,t)\) -- the velocity, \(\theta(x,t)\) -- the temperature, \(p(x,t)\) -- the pressure, \(\nu\), \(\eta\), \(k\) -- the viscosity, the volume expansion coefficient and the thermal conductivity, supposed constant, \(T_w(x)\) -- a given, non-constant, continuous function on \(\Gamma\), \(T_0=\min_{x\in\Gamma} T_w(x)\). The above boundary value problem is brought to a more simple form, (P), by means of a variable change, which is the base of the author's mathematical study. For the boundary problem (P), he proves the existence of a global, strong solution, its behaviour when \(t\to\infty\), the linearized operator constructed with the help of the steady solution \((t\to\infty)\), the global existence and the exponential decay of the solution. The results of the paper are to be considered as a continuation of the author's older results and as improvements of other authors' ones.
reviewed by: Vasile Ionescu
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76R05
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 861843
zbMATH Keywords: interior convection
zbMATH Keywords: Boussinesq approximate model
zbMATH Keywords: strong solution
zbMATH Keywords: steady solution
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1435
OpenAlex ID: W2052420077
DOI: 10.1006/JMAA.1995.1435
description: scientific article; zbMATH DE number 861843