Global existence and exponential stability of convection
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.
scientific article; zbMATH DE number 861843