Phase Transition in Non-isentropic Compressible Immiscible Two-Phase Flow with van der Waals Equation of State

This study establishes the global well-posedness of the compressible non-isentropic Navier-Stokes/Allen-Cahn system governed by the van der Waals equation of state $p(ρ,θ)=- aρ^2+\frac{Rθρ}{1-bρ}$ and degenerate thermal conductivity $κ(θ)=\tildeκθ^β$, where $p$, $ρ$ and $θ$ are the pressure, the density and the temperature of the flow respectively, and $a,b,R,\tildeκ$ are positive constants related to the physical properties of the flow. Navier-Stokes/Allen-Cahn system models immiscible two-phase flow with diffusive interfaces, where the non-monotonic pressure-density relationship in the van der Waals equation drives gas-liquid phase transitions. By developing a refined $L^2$-energy framework, we prove the existence and uniqueness of global strong solutions to the one-dimensional Cauchy problem for non-vacuum and finite-temperature initial data, without imposing smallness restrictions on the initial conditions. The findings demonstrate that despite non-monotonic pressure inducing substantial density fluctuations and triggering phase transitions, all physical quantities remain bounded over finite time intervals.