Uniqueness for the inhomogeneous incompressible Navier–Stokes–Cahn–Hilliard system with bounded density

Establish uniqueness of solutions for the inhomogeneous incompressible Navier–Stokes–Cahn–Hilliard system with bounded density.

Background

The paper identifies uniqueness as a major analytical difficulty for coupled incompressible multiphase fluid systems when the density is only bounded and lacks sufficient spatial regularity. In particular, the authors report that uniqueness for the inhomogeneous incompressible Navier–Stokes–Cahn–Hilliard system with bounded density had remained unresolved in the cited literature. The difficulty arises from the transport equation for the density, whose difference equation contains a term involving the velocity difference multiplied by the gradient of the second density; without additional regularity, this term cannot be controlled by the standard energy method and Gronwall’s inequality. The present paper addresses an analogous uniqueness challenge for the nonhomogeneous incompressible two-phase magnetohydrodynamic model by combining weighted energy estimates, integrability-shifting arguments, and a Lagrangian-coordinate formulation.