Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing

Technical Summary: Global Weak Solutions to a Compressible Navier–Stokes/Cahn–Hilliard System with Singular Entropy of Mixing

Problem Statement
This work addresses the mathematical analysis of a compressible Navier–Stokes/Cahn–Hilliard system modeling the evolution of a binary mixture of viscous fluids undergoing phase separation. The system is defined on a bounded domain $\Omega \subset \mathbb{R}^3$ over a time interval $(0, T)$. The governing equations describe the conservation of mass, momentum, and the evolution of the phase variable (mass concentration difference) coupled with a chemical potential.

The primary novelty and challenge of this study lie in the choice of the free energy density. Unlike previous literature on compressible mixtures, which typically employs regular potentials with polynomial growth, this work incorporates the physically relevant Flory-Huggins (logarithmic) entropy of mixing. This potential is singular at the boundaries of the physical interval $[-1, 1]$, enforcing the constraint that the phase variable $c$ must remain within $(-1, 1)$. The system is subject to no-slip boundary conditions for velocity and homogeneous Neumann conditions for the phase and chemical potential.

Methodology
The authors employ a compactness method based on a regularization-approximation scheme to prove the existence of global-in-time weak solutions for arbitrarily large initial data. The core methodology proceeds as follows:

1. Approximation Scheme: The singular Flory-Huggins potential $F(c)$ is approximated by a sequence of regular potentials $F_\varepsilon(c)$ defined on the entire real line, exhibiting at most quadratic growth at infinity. This allows the application of existing existence results for regularized systems (specifically those established by Abels and Feireisl for compressible mixtures with regular potentials).
2. Uniform Estimates: A critical component of the proof is the derivation of uniform bounds for the approximating solutions $(\varrho_\varepsilon, u_\varepsilon, c_\varepsilon)$ that are independent of the regularization parameter $\varepsilon$.
   • Energy Estimates: Standard energy inequalities provide bounds on the density, velocity, and gradient of the phase variable.
   • New Estimates for Density-Dependent Cahn-Hilliard: The authors derive novel estimates for the chemical potential $\mu_\varepsilon$ and the term $\varrho_\varepsilon F'_\varepsilon(c_\varepsilon)$ under minimal assumptions on the density (non-negative, $\gamma$-integrable with $\gamma > 3/2$). By exploiting the control of $\nabla_x \mu_\varepsilon$ from the energy balance and utilizing a specific test function involving the total mass difference, they establish uniform $L^2$ bounds on $\int_\Omega \varrho_\varepsilon |F'_\varepsilon(c_\varepsilon)| dx$.
   • Pressure Estimates: To handle the compressibility, the authors adapt the Lions-Feireisl theory. They derive improved integrability estimates for the pressure $p(\varrho_\varepsilon)$ by testing the momentum equation with a specific Bogovskii operator-based test function. This yields $L^q$ bounds on the pressure, where $q$ depends on the adiabatic exponent $\gamma$.
3. Compactness and Limit Passage: Using the uniform bounds, the authors extract convergent subsequences. Strong convergence of the density $\varrho_\varepsilon$ is established via the renormalized continuity equation and the effective viscous flux argument. Strong convergence of the phase variable $c_\varepsilon$ is obtained by adapting compactness arguments from previous incompressible studies, leveraging the strong convergence of the density and the uniform bounds on the chemical potential.

Key Contributions and Results
The main result, stated in Theorem 1.1, establishes the existence of global-in-time weak solutions to the system for $\gamma > 3/2$ in three-dimensional bounded domains. The solution satisfies the following properties:

• Regularity: The solution $(\varrho, u, c, \mu)$ possesses specific regularity classes, including $\varrho \in C_{weak}([0, T]; L^\gamma(\Omega))$ and $c \in L^\infty(0, T; W^{1,2}(\Omega)) \cap L^2(0, T; W^{2, \frac{2\gamma}{\gamma+1}}(\Omega))$.
• Physical Constraints: A significant achievement is the proof that the phase variable $c$ satisfies the physical bounds almost everywhere. Specifically:
  • $-1 \leq c \leq 1$ almost everywhere in $(0, T) \times \Omega$.
  • $-1 < c < 1$ almost everywhere on the set where the density is positive ($\{\varrho > 0\}$).
• General Initial Data: The result holds for initial data with finite total energy, without requiring the density to be strictly positive or bounded away from zero, nor requiring the initial concentration to be strictly within $(-1, 1)$ everywhere (only that the relative mass $M_c/M \in (-1, 1)$).

Significance and Claims
The paper claims to provide the first existence result for global weak solutions to the compressible Navier–Stokes/Cahn–Hilliard system with the physically relevant Flory-Huggins potential. The authors highlight that previous works on compressible mixtures were limited to regular potentials with polynomial growth, which fail to enforce the physical constraint $c \in [-1, 1]$.

The significance of the work lies in:

1. Bridging the Gap: It addresses the analytical gap between incompressible and compressible phase-field models by handling the singular entropy in the compressible regime.
2. Minimal Assumptions: The derivation of estimates for the density-dependent Cahn-Hilliard equation does not rely on the strict positivity or boundedness of the density, a condition often required in previous studies (e.g., in the incompressible case by the second author and Temam).
3. Physical Consistency: The proof rigorously demonstrates that the limit solution respects the physical bounds of the phase variable, ensuring the model's consistency with the underlying thermodynamics of phase separation.

The authors note that their approach relies on a compactness method and that the proof of the strong convergence of the densities follows established techniques (Lions-Feireisl theory) adapted to this specific coupled system. They also mention that the results can be extended to two-dimensional domains with improved regularity and that more general singular potentials satisfying similar growth conditions can be treated.