Global strong solution to a thermodynamic compressible diffuse interface model with temperature‐dependent heat conductivity in 1D

In this paper, we investigate the wellposedness of nonisentropic compressible Navier–Stokes/Allen–Cahn system with heat conductivity proportional to a positive power temperature. This describes flow two-phase immiscible heat-conducting viscous mixture. The phases are allowed shrink or grow due changes density in fluid and incorporates their transport current. We established global existence uniqueness strong solutions for 1D, which means no phase separation, vacuum, shock wave, mass, concentration will be developed finite time, although motion has large oscillations interaction between hydrodynamic phase-field effects is complex. Our result can regarded as natural generalization Kazhikhov–Shelukhin's (Kazhikhov Shelukhin, 1977) single-phase constant non-isentropic degenerate nonlinear conductivity.