Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system

TL;DR

This work analyzes a multi-species cross-diffusion Cahn–Hilliard system derived from a degenerate Ginzburg–Landau energy with a volume-filling constraint on a bounded domain. It develops a variational framework to characterize stationary states as minimizers of $E(\boldsymbol{u})$, proves interior uniform bounds and regularity, and reveals a decoupling that links the void species to a single-species energy; in convex regimes it obtains a unique constant minimizer and exponential convergence of the dynamics. The long-time behavior is quantified via a relative-energy approach, yielding explicit exponential rates that depend on cross-diffusion coefficients and domain constants, and the results extend to nonlocal/potential perturbations with minor adjustments. A structure-preserving two-point finite-volume scheme is introduced, ensuring mass conservation, positivity, and discrete energy dissipation, and numerical experiments in 1D and 2D illustrate spinodal decomposition, interface dynamics, and convergence to stationary states under varying parameter regimes.

Abstract

We study some properties of a multi-species degenerate Ginzburg-Landau energy and its relation to a cross-diffusion Cahn-Hilliard system. The model is motivated by multicomponent mixtures where crossdiffusion effects between the different species are taken into account, and where only one species does separate from the others. Using a comparison argument, we obtain strict bounds on the minimizers from which we can derive first-order optimality conditions, revealing a link with the single-species energy, and providing enough regularity to qualify the minimizers as stationary solutions of the evolution system. We also discuss convexity properties of the energy as well as long time asymptotics of the time-dependent problem. Lastly, we introduce a structure-preserving finite volume scheme for the time-dependent problem and present several numerical experiments in one and two spatial dimensions.

Figures (5)

• Figure 1: Profiles along time in the globally stable case. $\varepsilon=4,\beta=1,\kappa=k=1$.
• Figure 2: Profiles along time in the non-convex case. $\varepsilon=0.1,\beta=10,\kappa=k=1$.
• Figure 3: Profiles along time in the non-convex case. $\varepsilon=0.1, \beta=10,\kappa=1, k=2$.
• Figure 4: Residual of the Euler-Lagrange system in the case $k=1$ (left) and $k=2$ (right).
• Figure 5: Spinodal decomposition successively at times $t=0, 0.06,0.13,0.49,1.5$.

Theorems & Definitions (22)

• Lemma 2.1
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Lemma 3.1: Convexity of the free energy
• proof
• Corollary 3.2: Uniqueness of the minimizer
• Theorem 3.3: Global exponential stability of the constant steady state