In this paper, we study the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes/Allen-Cahn system, which describes the phase transitions in two-component patterns interacting with a compressible fluid. We establish the existence and space-time pointwise behaviors of global solutions to this non-conserved system. In order to control the source term consisting of the phase variable, we make use of the Green’s function and space-time weighted estimates to prove that the phase variable only contains the diffusion wave whose amplitude decays exponentially in time, so as to show that the density and momentum of the fluid obey the generalized Huygens’ principle.

In this paper, we investigate the wellposedness of the non-isentropic compressible Navier-Stokes/Allen-Cahn system with the heat-conductivity proportional to a positive power of the temperature. This system describes the flow 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 fluid and incorporates their transport with the current. We established the global existence and uniqueness of strong solutions for this system in 1-D, which means no phase separation, vacuum, shock wave, mass or heat or phase concentration will be developed in finite time, although the motion of the two-phase immiscible flow has large oscillations and the interaction between the hydrodynamic and phase-field effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov-Shelukhin’s result ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]) for the compressible single-phase flow with constant heat conductivity to the non-isentropic compressible immiscible two-phase flow with degenerate and nonlinear heat conductivity.