Global existence of solutions to a non-isentropic compressible diffuse interface model

In this paper, we investigate the existence of solutions to a class of compressible Navier–Stokes/Allen–Cahn systems describing the motion of a mixture of two viscous compressible fluids. Using the energy method and refined interpolation inequalities, we establish the global existence of solutions in the Sobolev spaces $H^i$ for $i=2,4$. The system features a temperature-dependent heat conductivity $k(\theta )={\theta }^{\beta }$ with $\beta >0$, which characterizes the flow of a two-phase immiscible, thermally viscous, compressible fluid mixture. The main technical difficulty of this work lies in the complicated estimates induced by high-order partial derivatives in the proof of solution regularity.