On the well posedness and large-time behavior for Boussinesq equations in Morrey spaces

Abstract

In this paper we are concerned with Boussinesq equations which model heat transport by natural convection inside a viscous incompressible fluid in $\mathbb{R} ^{n} $. We prove the well posedness of mild solutions and existence of self-similar ones in the framework of Morrey spaces. Our results allow us to consider singular and unbounded gravitational fields. We also study the long-time behavior of solutions and obtain existence of a basin of attraction for each self-similar solution. Moreover, conditions on initial data for solutions to vanish at infinity are given.