  In this paper we study the well-posedness and long-time dynamics of a
diffuse-interface model for the mixture of two viscous incompressible Newtonian
fluids with thermo-induced Marangoni effects. The governing system consists of
modified Navier--Stokes equations coupled with phase-field and energy transport
equations. We first derive an energy inequality that illustrates the
dissipative nature of the system under the assumption that the initial
temperature variation is properly small. Then we establish the existence of
weak/strong solutions via the energy method and discuss the long-time dynamics
as well as stability of the system.
