主动Cahn-Hilliard方程的数值分析与粗化动力学
Numerical analysis and coarsening dynamics of the Active Cahn-Hilliard equation
AI总结：
本文研究主动Cahn-Hilliard方程的分析与粗化动力学，提出新方法表征相分离构型，建立理论解释幂律转变，设计有限元数值格式并通过模拟验证结果。
AI中文摘要：
我们研究主动Cahn-Hilliard方程的分析及相序动力学，提供了超出当前研究水平的新结果，涉及该方程的适定性及其粗化动力学的表征。我们同时考虑正则多项式势和奇异对数势。具体而言，我们利用基于相平面异宿轨道的新方法，表征静态扭结轮廓与球形液滴态，得到与静态相分离构型相关关键量的精确值；此外，我们建立了一套理论，解释由活性和局部界面曲率驱动的表面张力修正，该理论可解释文献中推测的特征畴长因活性引发的幂律转变：$L(t)\sim t^{\frac{1}{z}}$，其中$z$从3变为4，表明在达到有限饱和长度前存在过渡效应。我们还设计了一种基于有限元的高效数值格式来近似该模型，证明其对正则势和奇异势均具有适定性与稳定性。在$d=2,3$维且采用奇异势的情况下，有限元近似的收敛分析证明了满足物理约束$\phi \in (-1,1)$的弱解在时间局部的存在性与唯一性；在$d=1$维且采用奇异势的情况下，我们在活性参数满足小条件时，建立了弱解的时间全局适定性与正则性。最后，我们针对不同测试案例开展数值模拟，结果表明该数值算法可正确复现预期的相分离动力学，且后期粗化动力学结果显示在达到后期长度饱和前，$z$从3到4的幂律转变，验证了我们的理论发现。
英文摘要：
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 $ϕ\in (-1, 1)$. 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.
