Numerical analysis and coarsening dynamics of the Active Cahn-Hilliard equation
We investigate the analysis and phase ordering dynamics of the active Cahn--Hilliard equation, providing novel results beyond the current state of the art concerning the well-posedness and the characterization of its coarsening dynamics. We consider both regular polynomial and singular logarithmic potentials. In particular, we exploit a new method based on heteroclinic trajectories in the phase plane to characterize static kink profiles and spherical droplet states, recovering the exact values of key quantities related to static phase-separated configurations; moreover, we develop a theory accounting for surface tension modifications driven by activity and local interface curvature, which explains the power-law shift $L(t)\sim t^{\frac{1}{z}}$ from $z=3$ to $z=4$ induced by activity for the characteristic domain length conjectured in the literature. This shows that there is a transitory effect before the attainment of a finite saturation length. We also design an efficient numerical scheme, based on finite elements, to approximate the model, proving its well-posedness and stability both for regular and singular potentials. In dimensions $d=2,3$ with singular potential, the convergence analysis of the finite element approximation proves the local-in-time existence and uniqueness of weak solutions satisfying the physical constraint $\phi \in (-1, 1)$. In dimension $d=1$ with singular potential, we establish global-in-time well-posedness and regularity of weak solutions under a smallness condition on the activity parameter. Finally, we show numerical simulations for different test cases which prove that our numerical algorithm correctly reproduces the expected phase separation dynamics. Moreover, we show the results for coarsening dynamics at late times which present a power law shift from $z=3$ to $z=4$ prior to reaching late-time length saturation, which confirms our theoretical findings.