Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system

1. Problem Statement

The paper investigates the mathematical well-posedness of the incompressible Cahn-Hilliard-Navier-Stokes (CHNS) system with variable density and anisotropic surface energy.

• Physical Context: The system models the hydrodynamic behavior of a two-phase fluid mixture where phase separation occurs. Unlike previous works focusing on isotropic surface tension (where energy depends only on the magnitude of the gradient), this study incorporates anisotropic surface energy, which is crucial for modeling materials with directional dependence (e.g., crystals or specific liquid crystals).
• Governing Equations: The system consists of:
  1. Mass Conservation: $\partial_t \rho + u \cdot \nabla \rho = 0$ (Transport equation for variable density).
  2. Phase Evolution: $\rho \partial_t \phi + \rho u \cdot \nabla \phi = \text{div}(D(\phi)\nabla \mu)$ (Cahn-Hilliard equation with convection).
  3. Momentum Conservation: $\rho \partial_t u + \rho(u \cdot \nabla)u - \text{div}(\nu(\phi)D(u)) = -\nabla \pi - \text{div}(\Gamma \xi(\nabla \phi) \otimes \nabla \phi)$ (Navier-Stokes with a Korteweg stress term).
  4. Incompressibility: $\text{div } u = 0$.
  5. Chemical Potential: $\rho \mu = \rho F'(\phi) - \text{div}(\Gamma \xi(\nabla \phi))$.
• Key Challenge: The presence of the anisotropic term $\Gamma(\nabla \phi)$, where $\Gamma$ is a homogeneous function of degree one, introduces nonlinearity and complexity in the stress tensor. The authors aim to prove the global existence of weak solutions in both 2D and 3D bounded domains, extending previous results limited to the isotropic case ($\Gamma(\nabla \phi) = |\nabla \phi|$).

2. Methodology

The proof relies on a Galerkin approximation scheme combined with a fixed-point argument and energy estimates. The methodology is structured in three main phases:

A. Regularization of the Potential

The logarithmic potential $F(\phi)$ (defined on $[-1, 1]$) is singular at the boundaries. The authors introduce a regularized potential $F_\epsilon$ (for $\epsilon > 0$) that is $C^2$ on $\mathbb{R}$ and approximates the logarithmic potential. This allows for the construction of approximate solutions without encountering singularities during the initial existence proof.

B. Galerkin Approximation and Local Existence

1. Finite Dimensional Subspaces: The solution is projected onto finite-dimensional subspaces $V_n$ (for velocity) and $H_n$ (for phase) generated by eigenfunctions of the Stokes and Laplace operators, respectively.
2. Linearization: A linearized version of the system is constructed. Using the Cauchy-Lipschitz theorem, a local solution $(\rho_n, u_n, \phi_n, \mu_n)$ is shown to exist on a small time interval $[0, T_0)$.
3. Fixed-Point Argument: A map is defined that takes an input velocity/phase field and returns the solution of the linearized system. The authors prove this map has a Schauder fixed point, ensuring the existence of a solution to the nonlinear approximate system.

C. Global Extension via Bihari's Inequality

A critical technical hurdle is extending the local solution to a global one ($T \to \infty$).

• Energy Balance: The authors derive a total energy balance equation involving kinetic energy, interfacial energy ($\int \Gamma^2(\nabla \phi)$), and free energy.
• Bihari's Inequality: To prevent the solution from blowing up in finite time, the authors apply Bihari's inequality (a generalization of Gronwall's inequality). By analyzing the differential inequality governing the energy $E(t)$, they prove that the energy remains bounded on a time interval that can be extended indefinitely. This step is crucial for establishing global existence for the approximate solutions.

D. Limit Passage ($n \to \infty$ and $\epsilon \to 0$)

1. Compactness: Uniform estimates (independent of $n$ and $\epsilon$) are derived for the approximate solutions in appropriate Sobolev and Besov spaces (e.g., $L^\infty(0,T; H^1)$ for $\phi$, $L^2(0,T; V)$ for $u$).
2. Strong Convergence: Using compactness arguments (Aubin-Lions lemma and properties of Besov spaces), strong convergence of the phase field $\phi_n$ and velocity $u_n$ is established.
3. Handling Anisotropy: A key difficulty is passing to the limit in the anisotropic stress term $\Gamma \xi(\nabla \phi) \otimes \nabla \phi$. The authors utilize the linearity of the map $p \mapsto \Gamma \xi(p)$ (Assumption H2) and the strong convergence of $\nabla \phi$ to show that this term converges strongly in $L^1$.
4. Removal of Regularization: Finally, by taking $\epsilon \to 0$, the solution to the regularized system converges to a weak solution of the original system with the logarithmic potential.

3. Key Contributions

• Extension to Anisotropy: This is the first work to rigorously prove the global existence of weak solutions for the anisotropic CHNS system with variable density in 3D. Previous results were largely restricted to the isotropic case.
• Variable Density: The analysis handles the coupling between the transport equation for density and the phase-field equation, which is more complex than the constant density case.
• Novel Use of Bihari's Inequality: The paper explicitly demonstrates how Bihari's inequality, combined with a fixed-point argument, can be used to extend local existence to global existence for this specific class of nonlinear PDEs, overcoming potential blow-up issues in the energy estimates.
• Handling Logarithmic Potentials: The rigorous treatment of the singular logarithmic potential via regularization and subsequent limit passage in the anisotropic setting.

4. Main Results

The paper establishes Theorem 2.1, which states that for initial data satisfying specific regularity and boundedness conditions (e.g., $\rho_0 \in L^\infty$, $u_0 \in L^2$, $\phi_0 \in H^1 \cap L^\infty$), there exists a global weak solution $(\rho, u, \phi, \mu)$ satisfying:

• Regularity:
  • $\rho \in C([0, T]; L^2) \cap L^\infty(\Omega \times (0, T))$.
  • $u \in L^2(0, T; V) \cap B^{1/4}_{2, \infty}(0, T; H)$.
  • $\phi \in L^\infty(0, T; H^1) \cap B^{1/4}_{\infty, \infty}(0, T; L^2)$.
  • $\mu \in L^2(0, T; H^1)$.
• Weak Formulation: The solution satisfies the integral formulations of the mass, momentum, phase, and chemical potential equations for all appropriate test functions.
• Energy Inequality: The solution satisfies the global energy dissipation law derived in Section 3.

5. Significance

• Theoretical Advancement: This work bridges a significant gap in the mathematical theory of diffuse-interface models. By moving from isotropic to anisotropic surface energy, it provides a rigorous foundation for simulating complex materials where surface tension depends on crystallographic orientation or direction.
• Physical Relevance: The results validate the mathematical models used in materials science for phenomena like solidification, crystal growth, and liquid crystal dynamics, where anisotropy is a dominant physical feature.
• Methodological Impact: The combination of Galerkin approximation, fixed-point theory, and Bihari's inequality offers a robust template for tackling other nonlinear coupled PDE systems where standard energy estimates are insufficient to guarantee global existence directly.

In summary, the paper successfully generalizes the existence theory for CHNS systems to the anisotropic regime, providing a rigorous mathematical framework for studying complex two-phase flows with direction-dependent surface properties.