Permeability parameter asymptotics in a Cahn--Hilliard system with third type transmission conditions

Technical Summary: Permeability Parameter Asymptotics in a Cahn–Hilliard System with Third Type Transmission Conditions

Problem Formulation
This paper investigates a coupled system of partial differential equations modeling phase separation and transitions, where the Cahn–Hilliard system governs the dynamics in a bulk domain $\Omega$ and an Allen–Cahn type equation governs the dynamics on the boundary $\Gamma = \partial\Omega$. The two domains are connected via a "third type" transmission condition characterized by a permeability parameter $K > 0$ (or equivalently $\alpha = 1/K$).

The specific system seeks functions $u$ (bulk order parameter), $\mu$ (chemical potential), $\xi$ (subgradient in the bulk), $\phi$ (boundary order parameter), and $\psi$ (subgradient on the boundary) satisfying:

1. Bulk Cahn–Hilliard: $\partial_t u - \Delta \mu = 0$ and $-\Delta u + \xi + \pi(u) - f = \mu$ in $Q = (0,T) \times \Omega$, with homogeneous Neumann condition $\partial_\nu \mu = 0$.
2. Boundary Allen–Cahn: $\partial_t \phi - \Delta_\Gamma \phi + \psi + \pi_\Gamma(\phi) + \frac{1}{K}(\phi - u) = f_\Gamma$ on $\Sigma = (0,T) \times \Gamma$.
3. Transmission Condition: $\partial_\nu u = \frac{1}{K}(\phi - u)$ on $\Sigma$.

Here, $\beta$ and $\beta_\Gamma$ represent possibly multivalued maximal monotone graphs (subdifferentials of convex potentials), and $\pi, \pi_\Gamma$ are Lipschitz continuous perturbations. The transmission condition allows for a discontinuity (gap) between the bulk trace and the boundary value, controlled by $K$.

Methodology
The authors employ a rigorous asymptotic analysis with respect to the permeability parameter $K$, investigating two distinct limiting regimes:

1. Zero Permeability ($K \to +\infty$, $\alpha \to 0$): Corresponds to a completely impermeable boundary where the bulk and boundary systems decouple.
2. Full Permeability ($K \to 0$, $\alpha \to +\infty$): Corresponds to a perfectly permeable boundary where the bulk and boundary phases are fully coupled, recovering the Dirichlet trace condition $u = \phi$.

The mathematical approach proceeds as follows:

• Approximation: The authors first establish the well-posedness of the intermediate problem (for fixed $K$) using the abstract theory of evolution equations governed by subdifferential operators. They utilize Yosida approximations ($\beta_\lambda, \beta_{\Gamma,\lambda}$) and a regularization parameter $\lambda$ to handle the multivalued nonlinearities and prove the existence of strong solutions.
• Uniform Estimates: A critical component of the methodology is the derivation of uniform estimates for the approximate solutions that are independent of the approximation parameter $\lambda$ and the permeability parameter $K$. These estimates explicitly track the dependence on $K$ (e.g., terms involving $1/\sqrt{K}$ or $1/K$) to ensure validity in the limits.
• Limiting Procedures: By passing to the limit $\lambda \to 0$ first to recover the solution for fixed $K$, and subsequently taking $K \to \infty$ or $K \to 0$, the authors utilize compactness arguments (Aubin–Lions theorems) and the demi-closedness of maximal monotone operators to identify the limit problems.
• Convergence Rates: The paper derives explicit rates of convergence for the solutions of the intermediate problem to the solutions of the limit problems.

Key Contributions and Results

1. Well-posedness of the Intermediate Problem: Theorem 2.1 establishes the existence and uniqueness of solutions for the coupled system with the third type transmission condition for any fixed $K > 0$, under general assumptions on the potentials (including singular and non-smooth cases).

2. Asymptotic Analysis for Zero Permeability ($K \to +\infty$):

   • Result: As $K \to +\infty$, the system decouples. The bulk solution $u$ converges to the solution of a standard Cahn–Hilliard system with homogeneous Neumann boundary conditions ($\partial_\nu u = 0$). The boundary solution $\phi$ converges to the solution of a standalone Allen–Cahn equation.
   • Convergence Rate: The authors prove that the convergence occurs with a rate of $O(\alpha^{1/2}) = O(K^{-1/2})$ in appropriate norms (Theorem 4.1).

3. Asymptotic Analysis for Full Permeability ($K \to 0$):

   • Result: As $K \to 0$, the transmission condition enforces the trace condition $u = \phi$ on $\Gamma$. The limit system is a Cahn–Hilliard system in the bulk coupled with an Allen–Cahn type dynamic boundary condition (specifically, $\partial_t \phi - \Delta_\Gamma \phi + \psi + \pi_\Gamma(\phi) + \partial_\nu u = f_\Gamma$). This recovers the model known in literature as the GMS model (or similar dynamic boundary condition models).
   • Technical Requirement: This limit requires additional assumptions (A7 and A8) regarding the compatibility of initial data ($u_0 = \phi_0$) and the growth relationship between the bulk and boundary potentials to ensure uniform bounds.
   • Convergence Rate: The paper establishes a convergence rate of $O(K^{1/2})$ for the solution variables and $O(K)$ for the difference between the bulk trace and boundary value (Theorem 4.2).

Significance and Claims
The paper positions itself within the context of transmission problems and dynamic boundary conditions, noting that dynamic boundary conditions can be viewed as the zero-thickness limit of transmission problems. The authors claim that their work provides a rigorous mathematical justification for these limits by explicitly analyzing the permeability parameter.

• Interpolation: The intermediate problem serves as a bridge between the "completely split" system (LW model, $K \to \infty$) and the "fully coupled" system (GMS model, $K \to 0$).
• Rigorous Justification: While the recovery of the Dirichlet condition ($u=\phi$) as $K \to 0$ is formally intuitive, the paper provides a rigorous proof of convergence and characterizes the resulting coupled equations, including the specific form of the dynamic boundary condition involving the normal derivative of the bulk variable.
• Quantitative Analysis: A significant aspect of the work is the derivation of explicit convergence rates ($O(K^{-1/2})$ and $O(K^{1/2})$), which quantifies how quickly the system transitions between the decoupled and coupled regimes.

The authors do not propose new physical applications or experimental setups but rather focus on the mathematical analysis of the asymptotic behavior of the governing equations, contributing to the theoretical understanding of phase separation models with dynamic boundaries.