[Paper Review] On the Cahn-Hilliard-Brinkman system

Stefano Bosia, Monica Conti+1 more|arXiv (Cornell University)|Feb 25, 2014

Solidification and crystal growth phenomena14 references4 citations

TL;DR

This paper establishes the well-posedness of the Cahn-Hilliard-Brinkman (CHB) system modeling phase separation in incompressible binary fluids within a porous medium, proving global existence and uniqueness of weak solutions, existence of a global attractor, convergence of trajectories to equilibrium via Łojasiewicz-Simon inequality, and convergence of CHB solutions to the Cahn-Hilliard-Hele-Shaw (CHHS) system as viscosity ν → 0 in two dimensions with explicit error estimates in H¹ norm.

ABSTRACT

We consider a diffuse interface model for phase separation of an isothermal incompressible binary fluid in a Brinkman porous medium. The coupled system consists of a convective Cahn-Hilliard equation for the phase field $ϕ$, i.e., the difference of the (relative) concentrations of the two phases, coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for the fluid velocity $\mathbf{u}$. This equation incorporates a diffuse interface surface force proportional to $ϕ abla μ$, where $μ$ is the so-called chemical potential. We analyze the well-posedness of the resulting Cahn-Hilliard-Brinkman (CHB) system for $(ϕ,\mathbf{u})$. Then we establish the existence of a global attractor and the convergence of a given (weak) solution to a single equilibrium via Łojasiewicz-Simon inequality. Furthermore, we study the behavior of the solutions as the viscosity goes to zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw (CHHS) system. We first prove the existence of a weak solution to the CHHS system as limit of CHB solutions. Then, in dimension two, we estimate the difference of the solutions to CHB and CHHS systems in terms of the viscosity constant appearing in CHB.

Motivation & Objective

• To establish the existence and uniqueness of weak solutions for the Cahn-Hilliard-Brinkman (CHB) system in a bounded domain with no-slip and Neumann boundary conditions.
• To prove the existence of a global attractor for the CHB system, demonstrating long-term dissipative behavior.
• To show that every weak solution converges to a single equilibrium state with an explicit convergence rate using the Łojasiewicz-Simon inequality.
• To analyze the limit of CHB solutions as viscosity ν → 0, proving convergence to a weak solution of the Cahn-Hilliard-Hele-Shaw (CHHS) system.
• To derive an explicit error estimate in the H¹ norm between strong solutions of CHB and CHHS systems in two spatial dimensions, showing convergence rate of order ν^{1/2}.

Proposed method

• Formal derivation of the CHB system as a coupled system: a convective Cahn-Hilliard equation for the phase field φ and a modified Brinkman equation for the velocity u, with a diffuse interface surface force proportional to φ∇μ.
• Use of variational formulation and Galerkin approximation to prove existence and uniqueness of weak solutions in H¹(Ω) × H¹₀(Ω) for φ and u.
• Application of energy estimates and a priori bounds in H²(Ω) and H¹(Ω) to control higher-order norms and ensure uniform boundedness.
• Employment of the Łojasiewicz-Simon inequality to establish convergence of trajectories to equilibrium with explicit decay rate.
• Use of compactness and weak convergence arguments to pass to the limit ν → 0 and prove existence of a weak solution to the CHHS system.
• Derivation of an L²(0,T;H¹) estimate for the difference between CHB and CHHS solutions via Gronwall’s lemma and uniform bounds, yielding O(ν^{1/2}) convergence in H¹ norm.

Experimental results

Research questions

• RQ1 Does the Cahn-Hilliard-Brinkman system admit a unique global weak solution for given initial and boundary data?
• RQ2 Does the CHB system possess a global attractor in the phase space, indicating long-term dissipative dynamics?
• RQ3 Can each weak solution of the CHB system be shown to converge to a single equilibrium state as t → ∞?
• RQ4 What is the behavior of CHB solutions as the viscosity ν approaches zero, and does the system converge to the Cahn-Hilliard-Hele-Shaw system?
• RQ5 Can the difference between strong solutions of CHB and CHHS systems be estimated in terms of ν, and what is the convergence rate?

Key findings

• The CHB system admits a unique global weak solution (φ, u) ∈ L∞(0,T;H¹) ∩ L²(0,T;H²) × L²(0,T;H¹₀) for any T > 0, with continuous dependence on initial data.
• The CHB system possesses a global attractor in the H¹ × H¹₀ phase space, indicating that all trajectories eventually enter and remain in a bounded absorbing set.
• Each weak solution of the CHB system converges to a single equilibrium state as t → ∞, with a convergence rate controlled by the Łojasiewicz-Simon inequality.
• As ν → 0, the CHB system converges to the CHHS system, and a global weak solution to the CHHS system exists as the limit of CHB solutions.
• In two dimensions, the difference between strong solutions of CHB and CHHS systems satisfies the estimate ‖φν − φ‖₁² ≤ C_T(‖φ₀^ν − φ₀‖₁² e^{C_T} + ν^{1/2}), implying O(ν^{1/2}) convergence in H¹ norm.
• The convergence in H² norm is not established due to the lack of strong continuity of the semigroup associated with the CHHS system.