Well-posedness and attractors for a 2D Boussinesq system with partial dissipation

Abstract

This article is devoted to the global well-posedness and the long-time behavior of solutions of a 2D Boussinesq equations with partial dissipation. We prove that this system is global well-posed under some weaker assumptions on the initial data and has a weak sigma-attractor which retains some of the common properties of global attractors for the dissipative dynamical system, moreover, the local attractor which is the composition of the weak sigma-attractor is upper semicontinuous under small viscosity perturbations.