Periodic solutions of the heat convection equations in exterior domains (Q1364458)

Let \(\Omega\) be the exterior of a compact set \(K\) in \(\mathbb{R}^3\) whose boundary, \(\Gamma\), is \(C^2\). Let \(\widehat\Omega= \Omega\times(0, \infty)\) and \(\widehat\Gamma= \Gamma\times(0,\infty)\). The author considers the problem of finding functions \(u= (u_1,u_2,u_3)\), \(p\), \(\theta\) of \((x,t)\in\widehat\Omega\) which satisfy the system \[ u_t+ (u\cdot\nabla)u= -(\nabla p)/\rho+ \nu\nabla u+\{1- \alpha(\theta-\Theta_0)\}g, \] \[ \text{div}\cdot u=0,\quad \theta_t+ (u\cdot\nabla)\theta= \kappa\Delta\theta \] in \(\widehat\Omega\), \(u\) vanishes on \(\widehat\Gamma\), \(\theta(x,t)= \chi(x,t)\) is given on \(\widehat\Gamma\), \(u\) and \(\theta\) tend to zero as \(|x|\to\infty\), and \(u(\cdot,0)= u(\cdot, T)\), \(\theta(\cdot, T)=\theta(\cdot, T)\). Here \(u\) is the velocity vector, \(p\) the pressure, \(\theta\) the temperature, \(\Theta_0\) a reference temperature, \(\nu\) the kineteic viscosity, \(\kappa\) thermal conductivity, \(\alpha\) the coefficient of volume expansion, \(\rho\) the density at \(\Theta_0\), and \(g= g(x)\) is the gravitational vector. It is assumed that \(g=O(|x|^{-5/2+\varepsilon})\) as \(|x|\to\infty\), \(\varepsilon>0\) arbitrary. The main theorem states that if \(|g|< \sqrt{\kappa\upsilon}/6^{4/3}\alpha\), a periodic solution exists, where \(|\cdot|\) is the \(L^{3/2}(\Omega)\) norm. The proof uses a method of extending domains and a priori estimates.