Nonhomogeneous Incompressible NSCH Systems

• The NSCH system is a diffuse-interface framework modeling binary two-phase flows with variable density and unmatched constituent properties.
• It distinguishes among AGG-type, transported-density, and quasi-incompressible formulations, each employing a different incompressibility mechanism.
• Analytical studies establish weak and strong well-posedness, energy dissipation laws, and numerical methods for addressing its nonlocal, high-order dynamics.

Searching arXiv for recent and foundational papers on the nonhomogeneous incompressible Navier–Stokes–Cahn–Hilliard system. The nonhomogeneous incompressible Navier–Stokes–Cahn–Hilliard system is a diffuse-interface framework for binary two-phase flow in which hydrodynamics, capillarity, and phase separation are coupled while the mixture is not homogeneous in density. In the standard variable-density interpretation, each constituent fluid is incompressible, but the constituent densities differ, so the total density varies with composition; in other formulations, the density is transported as an independent variable and enters the inertial, phase-transport, and chemical-potential laws. The terminology covers several mathematically distinct regimes: divergence-free volume-averaged models of Abels–Garcke–Grün type, transported-density models with $\operatorname{div}u=0$, and quasi-incompressible reductions in which the velocity divergence is generally not zero and the reduced momentum law becomes nonlocal (Abels et al., 2012, Freistuhler et al., 2013, Rui et al., 2024).

1. Terminology and model classes

A central point in the literature is that “nonhomogeneous incompressible” does not designate a single PDE. In one influential class, the velocity is the volume-averaged velocity and remains solenoidal even though the mixture density varies with the phase variable. In another class, the density is an independent transported unknown, again coupled to a divergence-free velocity. In a third class, incompressible constituents with different specific volumes lead, after elimination of the concentration, to a quasi-incompressible system in which $\nabla\!\cdot u$ is constrained by the Cahn–Hilliard mechanism rather than set to zero identically (Abels et al., 2012, Freistuhler et al., 2013).

A useful way to organize the subject is to distinguish the following formulations.

Formulation	Characteristic relation	Incompressibility mechanism
AGG-type unmatched-density model	$$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$	$\operatorname{div}v=0$ for the volume-averaged velocity
Transported-density model	$\rho_t+u\cdot\nabla\rho=0$	$\operatorname{div}u=0$, with density entering inertia and phase transport
Quasi-incompressible reduction	$\rho^{-1}=c\tau_1+(1-c)\tau_2$	$\nabla\!\cdot u$ determined through an elliptic relation tied to the chemical potential

The phrase also excludes an important family of nearby but different models: matched-density incompressible NSCH systems. Those keep the density fixed, often equal to $1$, and therefore are not nonhomogeneous in the density-dependent sense relevant here, even when viscosity is phase-dependent or the chemical potential is nonlocal (Keim et al., 2024).

2. Governing equations and thermodynamic structure

A canonical divergence-free nonhomogeneous formulation is the model attributed to Abels, Garcke, and Grün. In the version analyzed for degenerate mobility, the system reads

$$\partial_t(\rho v)+\operatorname{div}(\rho v\otimes v) -\operatorname{div}(2\eta(\varphi)Dv)+\nabla p +\operatorname{div}(v\otimes \beta J) = -\operatorname{div}\bigl(a(\varphi)\nabla\varphi\otimes\nabla\varphi\bigr),$$

$\nabla\!\cdot u$0

$\nabla\!\cdot u$1

$\nabla\!\cdot u$2

with

$\nabla\!\cdot u$3

The extra transport term $\nabla\!\cdot u$4 is the distinctive correction required by unmatched densities; it replaces the standard constant-density convection law and is part of the thermodynamically consistent structure (Abels et al., 2012).

In that same framework, the total energy is

$\nabla\!\cdot u$5

and weak solutions satisfy the dissipation inequality

$\nabla\!\cdot u$6

with $\nabla\!\cdot u$7. This formulation is particularly important when $\nabla\!\cdot u$8 degenerates at the pure phases, because then diffusion is confined to the interfacial region and $\nabla\!\cdot u$9 almost everywhere on $$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$0 (Abels et al., 2012).

A second major class is the transported-density model studied with Landau free energy: $$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$1

$$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$2

$$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$3

$$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$4

$$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$5

Here nonhomogeneity appears through the weighted inertia, the weighted phase transport, and the density-weighted constitutive law for $$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$6. The exact energy law is

$$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$7

which is the starting point for the strong-solution theory in three dimensions (Rui et al., 2024).

3. Reduction, nonlocality, and quasi-incompressibility

A different route begins from two incompressible constituents with distinct specific volumes $$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$8. In that setting,

$$\rho(\varphi)=\frac{\widetilde\rho_1+\widetilde\rho_2}{2}+\frac{\widetilde\rho_2-\widetilde\rho_1}{2}\varphi$$9

so the concentration is no longer independent: $\operatorname{div}v=0$0 This is the precise sense in which the mixture is “nonhomogeneous incompressible”: the constituents are incompressible, but the mixture density varies with composition (Freistuhler et al., 2013).

For the isothermal NSCH equations, elimination of $\operatorname{div}v=0$1 converts the Cahn–Hilliard equation into an elliptic relation driven by the velocity divergence: $\operatorname{div}v=0$2 If $\operatorname{div}v=0$3, then

$\operatorname{div}v=0$4

Substituting this into the pressure law yields a reduced momentum equation of Korteweg type with an additional nonlocal isotropic stress,

$\operatorname{div}v=0$5

The resulting reduced system is therefore not a local Navier–Stokes–Korteweg model but an integro-differential one (Freistuhler et al., 2013).

This reduction sharpens a recurring conceptual distinction. In AGG-type models the chosen velocity remains solenoidal, whereas in the quasi-incompressible reduction $\operatorname{div}v=0$6 is generally nonzero and is controlled by the elliptic inversion associated with the chemical potential. The nonlocal term $\operatorname{div}v=0$7 is the signature of Cahn–Hilliard diffusion after elimination of the concentration variable. On $\operatorname{div}v=0$8, $\operatorname{div}v=0$9 becomes convolution with the Poisson or Newton kernel, so the reduced NSCH dynamics is spatially nonlocal in a literal sense (Freistuhler et al., 2013).

4. Weak and strong well-posedness

For the AGG model with degenerate mobility

$\rho_t+u\cdot\nabla\rho=0$0

global-in-time weak solutions were proved in dimensions $\rho_t+u\cdot\nabla\rho=0$1. The solution concept uses the flux $\rho_t+u\cdot\nabla\rho=0$2 rather than $\rho_t+u\cdot\nabla\rho=0$3, because degeneracy destroys global $\rho_t+u\cdot\nabla\rho=0$4-control of the chemical-potential gradient. The theorem yields a weak solution $\rho_t+u\cdot\nabla\rho=0$5, the energy inequality, the physical bound $\rho_t+u\cdot\nabla\rho=0$6, and the interfacial property $\rho_t+u\cdot\nabla\rho=0$7 almost everywhere on $\rho_t+u\cdot\nabla\rho=0$8. No uniqueness is claimed (Abels et al., 2012).

In the transported-density Landau-potential model, local strong solutions are taken from the framework of Giorgini and Temam under the positivity assumption

$\rho_t+u\cdot\nabla\rho=0$9

For maximal existence time $\operatorname{div}u=0$0, the continuation criterion is Serrin-type and depends only on the velocity: $\operatorname{div}u=0$1 Under the smallness condition

$\operatorname{div}u=0$2

the strong solution is global and satisfies exponential decay of the natural energy. The proof combines the exact energy identity, weighted Poincaré estimates, elliptic control of $\operatorname{div}u=0$3 and $\operatorname{div}u=0$4, and a higher-order differential inequality for $\operatorname{div}u=0$5, $\operatorname{div}u=0$6, $\operatorname{div}u=0$7, and $\operatorname{div}u=0$8 (Rui et al., 2024).

A further extension replaces the Newtonian stress by the power-law constitutive law

$\operatorname{div}u=0$9

and allows vacuum in the initial density: $\rho^{-1}=c\tau_1+(1-c)\tau_2$0 In that setting there is global weak existence in a bounded three-dimensional domain and local strong existence on the periodic domain $\rho^{-1}=c\tau_1+(1-c)\tau_2$1. The chemical potential is written in weighted form,

$\rho^{-1}=c\tau_1+(1-c)\tau_2$2

precisely to remain meaningful when $\rho^{-1}=c\tau_1+(1-c)\tau_2$3. The proof relies on a semi-Galerkin scheme, weighted energy estimates, a weighted Poincaré-type inequality, and a monotonicity method for the non-Newtonian stress. The paper does not prove uniqueness of strong solutions (Li et al., 12 Nov 2025).

5. Constitutive, thermal, and geometric variants

The unmatched-density incompressible NSCH framework has been extended to non-isothermal two-phase flow with thermo-induced Marangoni effects. In that model,

$\rho^{-1}=c\tau_1+(1-c)\tau_2$4

and the system couples Navier–Stokes, convective Cahn–Hilliard, and convective heat equations with temperature-dependent viscosity, mobility, thermal diffusivity, and surface tension

$\rho^{-1}=c\tau_1+(1-c)\tau_2$5

Global weak solutions are established in two and three dimensions for a singular potential, while 2D uniqueness is proved only in the matched-density case under the additional assumptions

$\rho^{-1}=c\tau_1+(1-c)\tau_2$6

A decisive analytical change is that the formal energy law is no longer purely dissipative because Marangoni and buoyancy terms remain on the right-hand side (Chen et al., 7 Mar 2026).

In a distinct, geometric sense of nonhomogeneity, NSCH systems have also been studied in periodically perforated porous media with a rapidly oscillating anisotropic viscosity tensor, a non-conservative source term in the Cahn–Hilliard equation, and mixed no-slip/free-slip boundary conditions on the pore space. For each fixed $\rho^{-1}=c\tau_1+(1-c)\tau_2$7, weak solutions exist on the pore domain $\rho^{-1}=c\tau_1+(1-c)\tau_2$8, and periodic homogenization as $\rho^{-1}=c\tau_1+(1-c)\tau_2$9 yields two effective macroscopic regimes depending on the limit of the capillarity strength $\nabla\!\cdot u$0: a Stokes–Cahn–Hilliard limit when $\nabla\!\cdot u$1, and a Navier–Stokes–Cahn–Hilliard limit with nonlinear convection and phase advection when $\nabla\!\cdot u$2 (Chakrabortty et al., 24 Dec 2025).

These variants show that the nonhomogeneous incompressible NSCH class is not limited to one constitutive choice. Density contrast, degenerating or phase-dependent mobility, variable viscosity, non-Newtonian rheology, thermal coupling, source terms, and heterogeneous geometry all alter the analytical structure, often by changing the available energy law or the compactness mechanism.

6. Reformulations, contrasts, and current directions

A recurring research direction is to reduce the fourth-order and nonlocal structure to forms more amenable to numerical treatment. A recent example is a first-order hyperbolic–elliptic relaxation built from artificial compressibility for Navier–Stokes, a friction-type approximation for Cahn–Hilliard, and relaxation of a third-order capillarity term. That system is explicitly a matched-density model with constant density $\nabla\!\cdot u$3, constant viscosity, and constant mobility, so it is not a direct theory of the nonhomogeneous incompressible NSCH system. Its relevance is methodological: it shows how entropy structures and hyperbolic discretizations can be introduced for diffuse-interface flow once the incompressibility constraint and the fourth-order operator are replaced by lower-order approximations (Keim et al., 2024).

This contrast matters because several adjacent literatures use “incompressible NSCH” in ways that are not nonhomogeneous in the density sense. Matched-density systems with nonlocal chemical potential, or with nonhomogeneous Dirichlet boundary data for the velocity, remain analytically important but do not by themselves constitute a nonhomogeneous incompressible NSCH model unless density variation or an equivalent unmatched-density mechanism is present (Frigeri et al., 2013, Bag et al., 2022).

The present landscape therefore contains two persistent distinctions. First, “incompressible” may mean a solenoidal volume-averaged velocity, a divergence-free velocity coupled to a transported density, or incompressible constituents whose reduction produces a quasi-incompressible, nonlocal system. Second, the available well-posedness theory is highly model-dependent: weak existence is robust across many formulations, but uniqueness, global strong solvability in three dimensions, and thermodynamically exact dissipation laws remain sensitive to density assumptions, mobility degeneracy, constitutive complexity, and auxiliary couplings such as temperature or porous-medium damping. A plausible implication is that progress on the general theory will continue to proceed through carefully structured subclasses rather than through a single universal nonhomogeneous NSCH model.