Local Existence and Uniqueness for the 3D Compressible Navier-Stokes/Cahn-Hilliard System with Vacuum

Abstract

We establish local existence and uniqueness of strong solutions to the compressible Navier-Stokes/Cahn-Hilliard system in a bounded smooth domain in $\mathbb R^3$, allowing the initial density to vanish. The initial data satisfy suitable regularity and compatibility conditions for the momentum and phase equations. The main difficulty arises from the simultaneous degeneracy of the momentum equation and the density-weighted phase equations, which complicates the construction of compatible positive-density approximations and uniform initial estimates. We overcome this difficulty by introducing a time-independent residual into the chemical relation. The residual preserves the phase compatibility exactly, keeps the initial phase field unchanged, and vanishes in $H^1$ as the approximation parameter tends to zero. For each positive-density approximation, we construct a local strong solution using a semi-discrete Galerkin scheme that discretizes only the phase variables and retains the full transport and momentum equations. We then derive a priori estimates independent of the density lower bound, which yield a common lifespan and allow us to pass to the vacuum limit. Uniqueness is proved using a modified difference energy that accounts for the coupling between the density and the chemical potential. To our knowledge, this is the first local well-posedness result for the three-dimensional compressible Navier-Stokes/Cahn-Hilliard system with initial vacuum.