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We consider a thermodynamically consistent model for the evolution ofthermally conducting two-phase incompressible fluids. Complementing previousresults, we prove additional regularity properties of solutions in the casewhen the evolution takes place in the two-dimensional flat torus with periodicboundary conditions. Thanks to improved regularity, we can also proveuniqueness and characterize the long-time behavior of trajectories showingexistence of the global attractor in a suitable phase-space.

Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions