On weak solutions for the stationary Cahn-Hillard-Navier-Stokes equations with singular potential

Technical Summary: Weak Solutions for the Stationary Cahn-Hilliard-Navier-Stokes Equations with Singular Potential

Problem Statement
This paper investigates the existence of weak solutions for the stationary Navier–Stokes–Cahn–Hilliard system describing a three-dimensional, compressible, two-phase fluid mixture with a diffuse interface. The system is governed by the conservation of mass, momentum, and concentration, coupled with a chemical potential equation. A distinguishing feature of the model is the inclusion of a singular free energy density of the Flory–Huggins (logarithmic) type. This choice ensures that the mass fraction $c$ remains within the physical range $(-1, 1)$ and allows for the presence of vacuum states ($\rho = 0$).

The specific boundary value problem involves a bounded domain $\Omega \subset \mathbb{R}^3$ with Dirichlet conditions for velocity and Neumann conditions for concentration and chemical potential. The primary mathematical challenges arise from two sources:

1. The Stationary Setting: Unlike evolutionary problems, the stationary system lacks an a priori energy inequality ($E(t) \leq E(0)$) driven by time evolution, which typically provides the necessary control over the singular potential and the concentration field.
2. Singular Potential and Vacuum: The logarithmic singularity in the free energy prevents standard Taylor expansion regularizations, and the potential degeneracy of the density near vacuum states complicates the compactness arguments required for passing to the limit.

Methodology
The authors establish existence through a two-level limiting process applied to a carefully constructed approximate system.

1. Specialized Regularization of the Potential:
   To address the lack of $L^2$ bounds on $c$ and the failure of standard Taylor expansions, the authors introduce a specialized regularization $f^\delta_2(c)$ for the singular logarithmic term. Unlike previous approaches, this approximation is constructed such that the difference $f^\delta_2(c) - \frac{\theta_c}{2}c^2$ grows only linearly as $|c| \to \infty$. This specific construction is designed to cancel the quadratic growth induced by the anti-diffusion term, thereby restoring the necessary compactness and allowing for uniform estimates.

2. Two-Level Approximation Scheme:

   • Level 1 ($\epsilon$-limit): The authors first introduce an artificial viscosity and regularization parameters ($\epsilon$) to construct a regularized system. Using a fixed-point argument (Schaefer's Fixed Point Theorem), they prove the existence of strong solutions for this approximate system. Uniform estimates independent of $\epsilon$ are derived, allowing the limit $\epsilon \to 0$ to be taken, yielding a weak solution to an intermediate system with the regularized free energy.
   • Level 2 ($\delta$-limit): The second step involves removing the regularization of the potential ($\delta \to 0$). A major difficulty here is controlling the $L^{3/2}$ integrability of the density. The authors employ a specific choice of artificial pressure and a delicate interpolation argument. This argument exploits the structure of the regularized potential, specifically the observation that $\frac{\partial f^\delta_2}{\partial c} - \theta_c c = O(\ln \frac{1}{\delta})$ for $|c| > 1+\delta$. This allows them to close the estimates for the density norm and establish uniform bounds independent of $\delta$.

3. Strong Convergence and Physical Bounds:
   Passing to the limit $\delta \to 0$ requires proving the strong convergence of the density to handle the non-monotonic pressure term. Adapting techniques from the theory of steady compressible Navier–Stokes equations, the authors establish the strong convergence of the density. Furthermore, they prove that the limiting concentration $c$ satisfies the strict physical bounds $-1 < c < 1$ almost everywhere on the support of the density ($\{\rho > 0\}$), ensuring the singular term is well-defined.

Key Results
The main theorem (Theorem 1.1) establishes the existence of a weak solution $(\rho, u, \mu, c)$ to the stationary problem under the following conditions:

• Domain: A bounded, simply connected domain with $C^2$ boundary.
• Forces: External forces $g_1, g_2$ are bounded.
• Adiabatic Exponent ($\gamma$):
  • $\gamma > 3/2$ if the external force is irrotational ($\nabla \times g_1 = 0$).
  • $\gamma > 5/3$ if the external force is rotational ($\nabla \times g_1 \neq 0$).
• Mass Constraints: The total mass $m_1 > 0$ and relative mass $m_2$ satisfy $m_2 \in (-m_1, m_1)$.

The solution satisfies the renormalized continuity equation, the weak formulation of the momentum and concentration equations, and the energy inequality. Crucially, the concentration satisfies $-1 < c(x) < 1$ almost everywhere where $\rho(x) > 0$.

Significance and Claims
The authors claim this work provides the first existence result for weak solutions of the steady compressible Navier–Stokes–Cahn–Hilliard system that simultaneously incorporates:

1. A singular free energy potential (Flory–Huggins type), which is physically consistent for binary mixtures but mathematically challenging due to the lack of global bounds on the concentration.
2. The presence of vacuum states ($\rho = 0$), which introduces degeneracy in the equations.

The paper notes that while the range of admissible adiabatic exponents $\gamma$ might be extendable based on results for single-fluid steady compressible Navier–Stokes equations, the current range is limited by the difficulty of obtaining weighted estimates for the concentration near the boundary due to the singularity of the free energy. The work represents a significant step in bridging the gap between evolutionary models (where energy inequalities provide control) and stationary models with singular potentials.