Initial value problems for the heat convection equations in exterior domains (Q1281859)

This is a very interesting paper, concerning the problem of heat conduction the Boussinesq approximation, for an exterior domain in \(\mathbb{R}\). The considered system consists of the Navier-Stokes equation with the buoyancy term and the heat equation with a convection term. First, an existence theorem is proved for a sequence of interior domains. The Galerkin approximation, a priori estimates and the Artzela-Ascoli theorem are used in a classical way. Then the extending domain method is used to obtain the solution for the given exterior domain, which is approximated by a sequence of interior domains. For the bidimensional case \((n=2)\) a uniqueness theorem is given, with some assumptions on the buoyancy term, by using Gronwall's lemma. For a more regular solution, a uniqueness theorem is also obtained for \(n=3\).