Stationary solutions of the heat convection equations in exterior domains (Q1385016)

Let \(\Omega= K^c\subset\mathbb{R}^3\), where \(K\) is a compact set whose boundary \(\partial K\) is of class \(C^2\). We put \(\partial\Omega= \Gamma= \partial K\). Then we consider the stationary problem for the heat convection equation (HCE) in \(\Omega\): \[ (u\cdot\nabla)u= -(\nabla p)/\varrho+ \{1-\alpha(\theta- \Theta_0)\}g+ \nu\Delta u\quad\text{in }\Omega, \] \[ \text{div }u= 0,\quad (u\cdot\nabla)\theta= \kappa\Delta\theta\quad\text{in }\Omega, \] \[ u|_\Gamma= 0,\quad \theta|_\Gamma= \Theta_0>0,\quad \lim_{| x|\to\infty} u(x)= 0,\quad \lim_{| x|\to\infty} \theta(x)= 0. \] The purpose of the present paper is to show the existence of stationary weak solutions of (HCE) by using the ``the extending domain method''. Moreover, we also study the uniqueness of a weak solution.