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In this paper, we investigate the wellposedness of the non-isentropiccompressible Navier-Stokes/Allen-Cahn system with the heat-conductivityproportional to a positive power of the temperature. This system describes theflow of a two-phase immiscible heat-conducting viscous compressible mixture.The phases are allowed to shrink or grow due to changes of density in the fluidand incorporates their transport with the current. We established the globalexistence and uniqueness of strong solutions for this system in 1-D, whichmeans no phase separation, vacuum, shock wave, mass or heat or phaseconcentration will be developed in finite time, although the motion of thetwo-phase immiscible flow has large oscillations and the interaction betweenthe hydrodynamic and phase-field effects is complex. Our result can be regardedas a natural generalization of the Kazhikhov-Shelukhin's result([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]) for the compressiblesingle-phase flow with constant heat conductivity to the non-isentropiccompressible immiscible two-phase flow with degenerate and nonlinear heatconductivity.

Global Strong Solution to a Thermodynamic Compressible Diffuse Interface Model with Temperature Dependent Heat-conductivity in 1-D