The “do nothing” problem for Boussinesq fluids

In this note we establish the local existence and global uniqueness of strong solutions to the Boussinesq approximation for incompressible fluids in three-dimensional open channels considering the “do nothing” conditions prescribed at the in/outlets. The existence proof rests on estimates for the corresponding linear problems followed by a fixed point argument. The uniqueness is proved using the technique of Gronwall lemma.