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In this paper we study the well-posedness and long-time dynamics of adiffuse-interface model for the mixture of two viscous incompressible Newtonianfluids with thermo-induced Marangoni effects. The governing system consists ofmodified Navier--Stokes equations coupled with phase-field and energy transportequations. We first derive an energy inequality that illustrates thedissipative nature of the system under the assumption that the initialtemperature variation is properly small. Then we establish the existence ofweak/strong solutions via the energy method and discuss the long-time dynamicsas well as stability of the system.

Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced Marangoni effects