Well-posedness of the nonhomogeneous incompressible Navier-Stokes/Allen-Cahn system
Yinghua Li, Wenlin Ye
TL;DR
The paper provides a rigorous well-posedness theory for a diffuse-interface model coupling nonhomogeneous incompressible Navier–Stokes equations with Allen–Cahn diffusion in a bounded domain. Using Galerkin approximations, energy methods, and compactness, it establishes global weak solutions in 2D and 3D, local-in-time strong solutions in 3D, and global strong solutions in 2D; in 3D a small-data regime yields global strong solutions with exponential decay. A key contribution is the handling of density-dependent viscosity and phase-field diffusion, ensuring bounded density and phase-field confinement, along with a weak-strong uniqueness principle. These results provide a solid theoretical foundation for the NSAC diffuse-interface framework in multicomponent, density-variant flows and underpin robust long-time behavior in simulations. Overall, the work advances the mathematical understanding of coupled NS-AC systems with density contrasts and phase-field dynamics, highlighting how energy-dissipation structures control global regularity and stability.
Abstract
In this paper, we investigate a system coupled by nonhomogeneous incompressible Navier-Stokes equations and Allen-Cahn equations describing a diffuse interface for two-phase flow of viscous fluids with different densities in a bounded domain $Ω\subset\mathbb R^d (d=2, 3)$. The mobility is allowed to depend on phase variable but non-degenerate. We first prove the existence of global weak solutions to the initial boundary value problem in 2D and 3D cases. Then we obtain the existence of local in time strong solutions in 3D case as well as the global strong solutions in 2D case. Moreover, by imposing smallness conditions on the initial data, the 3D local in time strong solution is extended globally, with an exponential decay rate for perturbations. At last, we show the weak-strong uniqueness.