AI 中文总结
本文针对1-3维带乘性噪声的随机Cahn--Hilliard方程，结合谱Galerkin与向后Euler格式，推导了其全离散近似的一致时间强收敛率，证明了不变测度的存在唯一性，数值实验验证了理论结果。
AI 中文摘要
本文研究空间维数$d\in\{1,2,3\}$下带乘性噪声的随机Cahn--Hilliard方程全离散近似的一致时间强收敛率。所提格式结合了空间上的谱Galerkin方法与时间上的向后Euler格式。主要分析难点源于状态依赖的随机扰动、非线性项缺乏全局单调性结构，以及Cahn--Hilliard算子的四阶特性，这些特征使得三维情形下一致$L^{\infty}$矩估计的推导极具挑战性。对于连续方程，通过对$\\|u\\|^p$应用Itô公式并引入能量泛函$\mathcal{E}(u(t))$，本文推导了解的一致矩有界性；在全离散层面，本文发展了离散能量估计，并通过归纳法封闭所需的高阶矩界。基于这些正则性估计，本文推导了全离散格式的一致时间强收敛率，还证明了精确动力学与全离散数值动力学的不变测度的存在性与唯一性，数值实验结果证实了理论发现。
英文摘要
This paper investigates the uniform-in-time strong convergence rates of a fully discrete approximation for the stochastic Cahn--Hilliard equation driven by multiplicative noise in spatial dimensions $d\in\{1,2,3\}$. The proposed scheme combines a spectral Galerkin method in space with a backward Euler scheme in time. The main analytical difficulties arise from the state-dependent stochastic perturbation, the absence of a global monotonicity structure for the nonlinear term, and the fourth-order nature of the Cahn--Hilliard operator. In particular, these features make the derivation of uniform $L^{\infty}$-moment estimates highly nontrivial in three dimensions. For the continuous equation, by utilizing the Itô formula to $\|u\|^p$ and introducing the energy functional $\mathcal{E}(u(t))$, we derive the uniform moment boundedness of the solution. At the fully discrete level, we develop discrete energy estimates and close the required high-order moment bounds through an induction argument. Based on these regularity estimates, we deduce uniform-in-time strong convergence rates for the fully discrete scheme. Moreover, we prove the existence and uniqueness of invariant measures for both the exact dynamics and the fully discrete numerical dynamics. Numerical experiments are provided to confirm the theoretical findings.
